Polo's Schubert filtration conjecture for tensor products of Demazure modules

Let $G$ be a semisimple algebraic group with Borel subgroup $B$, Weyl group $W$, and dominant weights $\lambda,\mu\in P^+$. For $v,w\in W$, let $V_v(\lambda)$ and $V_w(\mu)$ be Demazure modules, and let a Schubert module mean a $B$-module of the form $H^0(S,\mathcal{L}_{-\nu})$ for a union $S$ of Schubert varieties. Polo's Schubert filtration conjecture. The tensor product

$$V_v(\lambda)\otimes V_w(\mu)$$

admits a filtration in which each successive quotient is isomorphic to a Schubert module. In particular, there exist $\nu^1,\ldots,\nu^k\in P^+$ and lower order ideals $\mathcal{I}^1,\ldots,\mathcal{I}^k\subseteq W$ such that

$$\operatorname{char}(\mathcal{B}_v(\lambda))\cdot\operatorname{char}(\mathcal{B}_w(\mu))=\sum_{i=1}^k c_i\kappa_{\nu^i,\mathcal{I}^i}$$

for some $c_i\in\mathbb{Z}_{>0}$. The conjecture proposes a structural refinement of excellent-filtration theory for Demazure modules; the source provides no evidence here that it has been resolved.