Travkin's asymptotic mirabolic bimodule conjecture

Let $\nu$ be a partition of $N$, let $c_\nu\subset\mathfrak S_N$ be the corresponding two-sided Kazhdan–Lusztig cell, and let $a_\nu$ be its $a$-function. For a pair of partitions $\nu\supset\theta$, let $c_{\nu\supset\theta\subset\nu}\subset RB$ be the corresponding bimodule Kazhdan–Lusztig cell. Let $J_{\nu\supset\theta\subset\nu}$ be the asymptotic bimodule with basis $\{t_{\tilde w}:\tilde w\in c_{\nu\supset\theta\subset\nu}\}$, and let $\operatorname{St}(\nu)$ denote the set of standard tableaux of shape $\nu$.

Asymptotic mirabolic bimodule conjecture. The based bimodule

$$J_{\nu\supset\theta\subset\nu},\ \{t_{\tilde w}:\tilde w\in c_{\nu\supset\theta\subset\nu}\}$$

is isomorphic to the based regular bimodule

$$\operatorname{Mat}_{\operatorname{St}(\nu)},\ \{e_{T_1,T_2}\},$$

with $t_{\tilde w}$ mapped to $e_{T_1,T_2}$, where $(T_1,T_2)$ are obtained from $\tilde w$ by mirabolic RSK.

This is the asymptotic-bimodule strengthening of the expected RSK description of bimodule cells. The preceding degree bound is itself conjectural in the source, and no resolution is supplied.