Twice now I’ve done the Daily Activity (both times on Daredevil). When I finish and return to LA, my staff is in my bags and I have to re-equip it. Today was Southsun Survival; I think the last one I did was Crab Toss. To be clear, this is happening every time I do the Activity, for a total of two times.
I have not tested this with non-Elites or with normal Thief weapons on me only.