Cellular-basis and W-graph isomorphism conjecture for Hecke algebras

Let $\mathcal H$ be the Hecke algebra in the paper, let $\Lambda$ index its modules, and for each $\lambda\in\Lambda$ suppose that a $W$-graph affords an $\mathcal H$-module $V^\lambda$ with generic specialization $V^\lambda_K\cong E_v^\lambda$. Let $W^\lambda$ denote the module arising from the cellular basis in Theorem~. Cellular-basis and $W$-graph conjecture. The cellular basis can be chosen so that

$$W^\lambda\cong V^\lambda$$

for every $\lambda\in\Lambda$. This would identify the cellular modules with modules afforded by the explicitly given $W$-graphs; the surrounding text says that the authors had not proved the assertion, while related computational work solved the cited subproblem.