Did Imam Al-Hussein have a daughter named “Ruqayyah” or “Sukainah” who passed away in Damascus while she was around 3-4 years old?

Question
Did Imam Al-Hussein have a daughter named “Ruqayyah” or “Sukainah” who passed away in Damascus while she was around 3-4 years old? As far as I know, it is not cited clearly anywhere in acceptable first-hand references (e.g. al-Irshad). Even the first narrator of the story, Imaduddin al-Tabari, the author of “Kamel Baha’i”, left it ambiguous by not mentioning the little girl’s name [assuming that the narration per se is authentic; which might not be, due to lack of the chain of narrators (isnad)]. On the other hand, according to references such as “Tabaqat al-Kubra”, etc., “Sukainah bint Al-Hussein” had the chance of living long enough to raise her own children! I was wondering whether you think there is enough evidence supporting the existence of such a girl and the related story.

Concise answer
Even though many historians have not made mention of a small girl of Imam Hussein with the name of Ruqayyahh, Fatimah  Sughraa or any other name in their books, but some books do cite the story of this small girl and the tragic incidents that took place for her in the ruins of Sham (current day Syria).
There are also clues in our hadith and history sources that support her existence and we will mention a few as examples:
First: When Lady Zeynab was speaking to the head of her brother, Imam Hussein, she recited poems that consisted of the following sentence: "O Brother! Speak to little Fatimah for it is close for her heart to become empty (it is close for her to pass away)."
Second: When encountering Shimr at the final moments of his life, Imam Hussein states: "O my Zeynab! O my Sukainah! O My children! Who will be your guardian after me? O my Ruqayyahh! O my Umme Kulthoom! You are Allah's trust.  Today we have come close to the promised moment."
Shiekh Mufid says: "Sukainah is among the daughters of Imam Hussein and her mother's name is Rubab."
Sheikh Tabarsi says: "Sukainah, the daughter of Imam Hussein, was ten years old on the day of Ashura."
Therefore, many books have mentioned Sukainah as one of Imam Hussein's daughters.
It is understood from these ahadith that the Imam had another daughter also named Sukainah who at that time- before the incident of Karbala-had reached the age of marriage.
Taking what was said into consideration, one can conclude that a daughter of Imam Hussein (who was either named Ruqayyahh or Fatimah) passed away beside the head of her father in the ruins of Sham, and that this little girl was not Sukainah, the other daughter of the Imam who lived for many years after her father's martyrdom.

Detailed Answer
Many historians and those who write maqtals (accounts of what took place on Ashura) have mentioned two girls with the names of Fatimah  and Sukainah when counting the daughters of Imam Hussein.[1] Also, some have added the name of "Zeynab"[2], while other books have narrated the story of the Imam's small daughter and the tragic incidents that took place for her in the ruins of Sham.[3] Most of these writers have cited this story from the book of "Kamel Baha’i" which was written in seventh century A. H.
There are also clues in our hadith and history sources that support her existence and as an example we will mention one of them:
First: When Lady Zeynab  was speaking to the head of her brother, Imam Hussein, she recited poems that consisted of the following sentence: "O Brother! Speak to little Fatimah for it is close for her heart to become empty (it is close for her to pass away)."[4] This sentence indicates that such a small girl existed and could not bear the separation from her father.
For more information you can refer to: Question 7235 (website 7318) Ruqqayah, the daughter of Imam Hussein
When examining historical books and the Maqatel we find that Shia and Sunni historians have mentioned a daughter named Sukainah for Imam Hussein. In this regard Sheikh Mufid says:" Sukainah is among the daughters of Imam Hussein and her mother's name is Rubab."[5] Sheikh Tabarsi also mentions her name saying that: "Imam Hussein married her to Abdullah Bin Al Hassan - her cousin - that was martyred on the day of Ashura." [6] It is mentioned in the book Maghtal al-Hussein that: "She (Sukainah) married her cousin (Abdullah Bin Al Hassan) who was martyred on the day of Ashura before consummating their marriage and no child was born from this marriage."[7] Tabarsi also narrates that: "Sukainah, the daughter of Imam Hussein was 10 years old during the incident of Ashura."[8]
Dhahabi has also mentioned in his "Tarikh al-Islam" that Sukainah was the daughter of Imam Hussein and names up to 20 books that have mentioned Sukainah as the Imam's daughter.[9] Therefore many books have mentioned Sukainah as the daughter of Imam Hussein.  In addition to the books mentioned in this article, we have mentioned the names of the rest of these books in the footnotes.[10]
However, regarding her age, there is no clear narration, and it is only understood from the different opinions that during Ashura she was either married or she had reached the age of marriage.[11]  
Taking what was said in to consideration, one can conclude that a daughter of Imam Hussein (who was either named Ruqayyahh or Fatimah ) passed away beside the head of her father in the ruins of Sham, and that this little girl was not Sukainah, the other daughter of the Imam who lived for many years after her father's martyrdom.  

References
[1] Mufid, Muhammad bin Nu’man, Al-Irshaad, The Collection of the Works of Sheikh Mufid, vol. 2, pg. 135, Dar al-Mufid, Beirut, 1414 AH.
Ibn Shahr Ashub, Manaaqib Aal Abi Taleb, vol. 4, pg. 77, Allamah Publications Institute, Qom Seminary Print.
Tabarsi, A’laam al-Waraa, vol. 1, pg. 478, Aal Bayt Institute, first print, 1417 AH.
Mus’ab al-Zubayri, Nisab al-Quraysh, pg. 59, Dar al-Ma’aarif, Cairo, third print.
Beladheri, Ansaab al-Ashraaf, vol. 3, pg. 1288, Dar al-Fikr, Beirut , first print, 1401 AH.
Sibt bin Jowzi, Tadhkirah al-Khawass, pg. 349, Ahlul-Bayt Institute, Beirut, first print, 1401
[2] Al-Arbali, Kashf al-Ghummah fi Ma’rifat al-A’immah, vol. 2, pg. 38, researched by Rasuli, Tabriz, Suuq Masjid Jame’.
[3] Qummi, Sheikh Abbas, Nafas al-Mahmum, pp.415-416, Maktabah al-Heydariyyah Publications, first print, 1379.
Al-Iqaad, Shah Abdul-Azimi, pg. 179, researched by Razavi, Firuzabadi Publications, first print, 1411 AH.
Ha’eri, Ma’aali al-Sibtayn, vol.2, pg. 170, Al-Numaan Publications, Beirut, 1412 AH.
Qummi, Sheikh Abbas, Muntaha al-Aamaal, vol. 1, pg. 807, Hijrah Publications Institute, fourth print, 1411 AH.
Tabari, Imaduddin, Kamel Baha’i, vol. 2, pg. 179, Maktabah Al-Mustafavi.
[4] Majlisi, Muhamamad Baqir, Bihar al-Anwar, vol. 45, pg. 115 “…یا اخی فاطم الصغیرة کلمّا فقد کاد قلبها ان یذوبا”;
Al-Qanduzi, Yanabi’ al-Mawaddah, vol. 2, pg. 421, Al-Sharif Al-Radhiyy Publications, first print, 1371.
[5] Mufid, Muhammad bin Nu’maan, Al-Irshad, vol. 2, pg. 37, Ilmiyyah Islamiyyah Publications.
[6] Tabarsi, A’laam al-Waraa, vol. 1, pg. 418, Aal al-Bayt Publications; Mufid, Muhammad bin Nu’man, Al-Irshad, pg. 25; Arbali, Kashf al-Ghummah, pg. 157.
[7] Musawi, Abd al-Razzaaq, Maqtal al-Husein, pg. 397, Basirati Publications.
[8] Ibid.
[9] Al-Dhahabi, Tarikh al-Islam, vol. 7, pg. 371, Dar al-Kutub al-Arabi, Beirut, Lebanon.
[10] Abu al-Faraj al-Isfahani, Maqaatil al-Talebin, pp. 94, 119, 133, 167; Al-Baladheri, Ansaab al-Ashraaf, vol. 3, pg. 362; Ibn Hanaan, Al-Thuqaat, vol. 4, pg. 351, Al-Kutub al-Thaqaafiyyah Publications; Al-Bukhari, Al-Taarikh al-Saghir, vol. 1, pg. 273, Dar al-Ma’rifah, Beirut, Lebanon; Al-Usfuri, Taarikh Khalifah bin Khayaat, pg. 274, Dar al-Fikr, Beirut; Muhammad bin Sa’d, Al-Tabaqaat al-Kubra, vol. 8, pg. 475, Beirut; Al-Muzni, Tahdhib al-Kamaal, vol. 6, pg. 397, Al-Risalah Publications; Ibn Aamir, Taarikh al-Madinah, vol. 2, pg. 52, and vol. 29, pg. 69 in different pages, Damascus, Dar al-Fikr; Ibn Maakulaa, Ikmaal al-Kamaal, vol. 4, pg. 316 and vol. 7, pg. 107, Dar al-Kitab al-Islamiyy, Cairo; Majlisi, Muhammad Baqir, Bihar al-Anwar, vol. 45, pg. 169, pg. 47, Beirut; Qummi, Sheikh Abbas, Muntaha al-Aamaal, vol. 1, pg. 547, Huseini Publications.
[11] Mufid, Muhammad bin Nu’maan, Al-Irshad, vol. 2, pg. 22, Farsi translation, Rasuli Mahallati, Ilmiyyah Islamiyyah Publications