Travkin's character-sheaf formula conjecture for mirabolic Kazhdan–Lusztig basis elements

Let $\mathcal R$ be the mirabolic bimodule, let $\mathcal H$ be the Iwahori–Hecke algebra, let $\mathsf{CH}$ be the character-sheaf functor, and let $\tilde{\underline H}_{\tilde w}$ be a Kazhdan–Lusztig basis element. Let $[\mathcal F_{\lambda,\mu}]$ denote the classes of unipotent mirabolic character sheaves, with $|\lambda|+|\mu|=N$. For a pair $(\nu,\theta)$, let $\Upsilon(\lambda,\mu)=(\nu,\theta)$, and let $V^*_{\nu}\otimes V_{\nu}$ be the corresponding summand in the stated decomposition of $\mathcal R$.

Mirabolic character-sheaf formula conjecture.

$$\mathsf{CH}\tilde{\underline H}_{\tilde w}=\sum_{|\lambda|+|\mu|=N} f_{\lambda,\mu}(\tilde{\underline H}_{\tilde w})[\mathcal F_{\lambda,\mu}],$$

where $f_{\lambda,\mu}:\mathcal R\to\mathbb Z[\mathbf v,\mathbf v^{-1}]$ satisfies $f_{\lambda,\mu}(hr)=f_{\lambda,\mu}(rh)$ for all $r\in\mathcal R$ and $h\in\mathcal H$, and vanishes on every summand except $V^*_{\nu}\otimes V_{\nu}$ corresponding to $(\tilde\nu,\tilde\theta,\tilde\nu)\in\mathbf T$ with $\Upsilon(\lambda,\mu)=(\nu,\theta)$.

The conjecture predicts the character-sheaf expansion of the transformed Kazhdan–Lusztig basis. The source supplies no resolution status or further evidence, so it remains open.