Reducedness conjecture for Kazhdan–Lusztig ideal initial schemes

Let $v,w\in S_n$, let $I_{v,w}$ be the associated Kazhdan–Lusztig ideal, and let $\prec_{v,w,\pi}$ be one of the term orders determined by a generalized antidiagonal permutation $\pi$. The initial ideal ${\rm init}_{\prec_{v,w,\pi}}I_{v,w}$ defines a scheme. Reducedness conjecture for Kazhdan–Lusztig ideals. For some $\pi$, ${\rm init}_{\prec_{v,w,\pi}}I_{v,w}$ defines a reduced and equidimensional scheme, so the hypothesis and consequently the conclusion of Theorem~ (V) hold. This conjecture is presented as a complement to the paper's main theorem and is part of the proposed degenerations of Kazhdan–Lusztig ideals; the supplied text gives no resolution.