The graded-module character conjecture for canonical basis elements

Let $R$ be a reduced root system with equal parameters $t_s=t_\ell=:t$. For a weight $\lambda$, let $C'_{\lambda}$ be the canonical basis element and let $P_{v_\mu,v_\lambda}(t)$ be the corresponding Kazhdan–Lusztig polynomial. Graded-module character conjecture. For any weight $\lambda$, the polynomial $C'_{\lambda}$ is the $T$-character of a graded $B$-module. In particular, the positive integers $P_{v_\mu,v_\lambda}(1)$ represent weight multiplicities in $B$-modules. This is presented as a computationally motivated expected property beyond the anti-dominant case, where the canonical basis elements have a Weyl-character interpretation.