Perfect bases for Schubert cells

Let $G^\vee$ be the dual group with Weyl group $W$, and for $w\in W$ let $X_w^\vee=B^\vee\backslash B^\vee wB^\vee$ be the Schubert cell. Then:

Schubert-cell perfect-basis conjecture. For every $w\in W$, the coordinate algebra $\mathbb{C}[X_w^\vee]$ has a perfect basis.

The conjecture is known in the parabolic case $w={w_P}^{-1}$ by the proposition cited in the source. In general, the source says it would yield an associated crystal $\mathcal{B}(\mathbb{C}[X_w^\vee])$.