We discuss the problem of which subshifts support strictly asymptotically nilpotent CA, that is, asymptotically nilpotent CA which are not nilpotent. The author talked about this problem in AUTOMATA and JAC 2012, and this paper discusses the (lack of) progress since. While the problem was already solved in 2012 on a large class of multidimensional SFTs, the full solutions are not known for one-dimensional sofics, multidimensional SFTs, and full shifts on general groups. We believe all of these questions are interesting in their own way, and discuss them in some detail, along with some context.