Parabolic subgroup invariance conjecture for reduced reflection expressions

Let $W$ be a Coxeter group with reflection set $T$, and let ${\mathrm{Red}_T}(w)$ denote the set of minimal-length expressions of $w$ as a product of reflections. Suppose that

$$(t_1,t_2,\dots,t_k)\in {\mathrm{Red}_T}(w)$$

and that $W':=\langle t_1,t_2,\dots,t_k\rangle$ is parabolic. Parabolic subgroup invariance conjecture. For every

$$(q_1,q_2,\dots,q_k)\in {\mathrm{Red}_T}(w),$$

we have

$$W'=\langle q_1,q_2,\dots,q_k\rangle.$$

The conjecture asserts that whenever one reduced reflection expression generates a parabolic subgroup, the subgroup generated by the reflections in every reduced expression of the same element is identical. The surrounding results establish that parabolic subgroups preserve reflection length under passage to their canonical reflection sets; whether this stronger invariance statement holds in general is left open here.