Crystal realization of Schubert cells

Let $G^\vee$ be the dual group with Weyl group $W$, and let $X_{w^{-1}}^\vee$ denote the Schubert cell indexed by $w^{-1}$. Let $\mathcal{B}(\mathbb{C}[X_{w^{-1}}^\vee])$ be its crystal basis, and let $\mathcal{B}_{f_w,\Theta^-_w}$ be the upper normal crystal associated with $w$.

Schubert-cell crystal conjecture. For every $w\in W$,

$$\mathcal{B}(\mathbb{C}[X_{w^{-1}}^\vee])\cong \mathcal{B}_{f_w,\Theta^-_w}.$$

This is presented as a conjectural refinement of the perfect-basis conjecture for Schubert cells. The preceding character computation establishes equality of characters, while the isomorphism itself remains conjectural in the source.