Lusztig's relative Kazhdan–Lusztig cell conjecture

Let $W$ be a Coxeter group with weight function $L$, let $W_I\subseteq W$ be a parabolic subgroup, and let $Y_I$ denote the corresponding set of distinguished coset representatives. For $u,v\in W_I$ and $x,y\in Y_I$, consider the relative Kazhdan–Lusztig relations $\leqslant_{{\mathcal{L}},I}$, $\sim_{{\mathcal{LR}},I}$, and $\sim_{{\mathcal{L}},I}$. Lusztig's relative cell conjecture. One has the implication

$$ux \leqslant_{{\mathcal{L}},I} vy \quad\text{and}\quad u \sim_{{\mathcal{LR}},I} v \quad\Rightarrow\quad u \sim_{{\mathcal{L}},I} v \quad\text{and}\quad x=y.$$

When $W_I=W$, this reduces to Lusztig's original assertion that $u\leqslant_{{\mathcal{L}}}v$ and $u\sim_{{\mathcal{LR}}}v$ imply $u\sim_{{\mathcal{L}}}v$. The original assertion is known in the equal-parameter case for finite and affine Weyl groups, as well as in several other cases, but the relative version is conjectured for all $W$, $L$, and choices of $W_I\subseteq W$.