$J$-ascent sets in parabolic quotients of Coxeter groups

Let $(W,S)$ be a Coxeter system and let $J\subseteq S$. The right $J$-ascent set $$ A_R^J(w)=\{s\in S\mid ws\in{}^JW,\ \ell(ws)=\ell(w)+1\} $$ is a natural refinement of the classical ascent set adapted to the parabolic quotient ${}^JW$. We establish a local transition formula describing the behavio