My thoughts are - yes. This is pure speculation on my part, but the dates are quite specific on the possession warrant, per this news article. I am assuming that LE has evidence to show that AT purchased or took possession of a firearm on September 1st.
As for no longer possessing the firearm after November 20th, I believe LE must have something specific that shows evidence of this. Do we know when LE identified AT and BB as POI's and when contact was made? In the one video, AT stated that the Lincoln police (I believe) called "Katie" who was BB's supposed sometime gf and told her she was in danger, so AT and BB took Katie home.
I don't believe that Katie exists, however, I do think the Lincoln police were in contact with AT and BB and were asking a LOT of questions about the whereabouts of SL. If the contacts with LE began within days of SL's disappearance, I imagine AT and BB were quite anxious to get rid of the firearm, especially if the firearm was used to murder SL.
If AT disposed of the firearm, how would LE know, and in particular, how would they know the date? The only thing I can think of is there might be an eyewitness who saw him get rid of the firearm. I wouldn't be surprised at all if AT and BB were under surveillance after LE identified them as POI's.