Vertex-decomposable degeneration conjecture for Kazhdan–Lusztig ideals

Let $v,w\in S_n$, choose a generalized antidiagonal permutation $\pi$ satisfying the reducedness conjecture, and let $\Delta_{v,w}$ be the Stanley–Reisner simplicial complex associated to ${\rm init}_{\prec_{v,w,\pi}}{\mathcal N}_{v,w}$. Vertex-decomposable degeneration conjecture. There exists a choice of $\pi$ among those satisfying the reducedness conjecture such that $\Delta_{v,w}$ is homeomorphic to a vertex decomposable, and hence shellable, ball or sphere; in particular, the hypothesis of Theorem~ (VI) holds. The conjecture proposes a particularly well-behaved combinatorial degeneration: vertex decomposability implies shellability and Cohen–Macaulayness, but the supplied text gives no resolution.