Walaikumussalam
In this issue there are the differences but the majority of the historian they believe that Ahlul bait as they spent one year the prison of Kufa and Damishq(Damascus)
And first Zair of Imam Husain Hazrat JABIR as he was bless to do the ziyarat of Imam Husain as in second Arbaeen of Imam Husain as where he met Imam Zainul abedin as and paid his condolences to him, so these things prove that it was 20th Safar 62 ah they reached Karbala.
Wassalam