In general, if I understand correctly, the representation theory of the braid groups is quite complicated, and there's no classification of the irreducibles. However, the braid groups form a sort of system of groups just as the symmetric groups do, and so one can ask about representation stability for coherent systems of representations of braid groups. One example might be the sequence of homologies of configuration spaces. Is anything at all known about when, if ever, representation stability occurs in this setting? For instance, is it known perhaps that the sequence of homologies of configuration spaces cannot exhibit representation stability? This is a very basic question, mostly just a reference request.

Note that I am not asking about representation stability for the braid groups themselves, which is known, nor am I asking about representation stability for the sequence of homologies of configuration spaces as symmetric group representations, which also is known.