The inverse Kazhdan–Lusztig matrix conjecture for $\mathfrak{gl}(m/n)$

Let $\mu$ be integral dominant and $r$-fold atypical with respect to the roots $\gamma_1<\gamma_2<\cdots<\gamma_r$ ($\gamma_i\in\Delta_{1,+}$). Let $\lambda_\theta$, for $\theta=(\theta_1,\ldots,\theta_r)\in\{0,1\}^r$, be the $2^r$ integral dominant weights determined by the composition-factor multiplicity conjecture. Define

$$a_{\lambda_\theta,\mu}(q)=(-q)^{|\theta|},\qquad |\theta|=\sum_i\theta_i,$$

and set $a_{\lambda,\mu}(q)=0$ for all other $\lambda$. The inverse Kazhdan–Lusztig matrix conjecture. The inverse of the triangular matrix $A_q=(a_{\lambda,\mu}(q))$ is the matrix $K_q=(K_{\lambda,\mu}(q))$ of Kazhdan–Lusztig polynomials.

This conjecture proposes a $q$-analogue of the composition-factor multiplicity matrix and would provide a direct determination of the Kazhdan–Lusztig polynomials for $\mathfrak{gl}(m/n)$. The supplied material gives no evidence that it has been resolved.