Parabolic maximality conjecture for standard homogeneity of Kazhdan–Lusztig ideals

Let $v,w$ be permutations with $v\leq w$, and let $I_{v,w}$ denote the Kazhdan–Lusztig ideal. Suppose that $I_{v,w}$ is standardly homogeneous. If $ws_i<w$ but $vs_i>v$, then $I_{vs_i,w}$ is standardly homogeneous; similarly, if $s_iw<w$ but $s_iv>v$, then $I_{s_iv,w}$ is standardly homogeneous. Let $v_{\rm \max}$ be the maximal element in the relevant parabolic double coset $S_{\mathcal T}vS_{\mathcal T'}$. Parabolic maximality conjecture. Under these hypotheses, $I_{v_{\rm \max},w}$ is standardly homogeneous. This would reduce the computation of all multiplicities to the parabolically maximal cases, where standard homogeneity can be checked using the stated multiplicity-computation facts.