The Hall–Littlewood Schur-expansion conjecture via an injective Yamanouchi map

Let $b4$ be a diagram with $n$ cells, let $f_b4$ be an injective function as in the injective Yamanouchi-word conjecture, and let $c0$ range over standard Yamanouchi words of partition shape $bb$. Let $c1am_b4$ and $bcj_b4$ denote the inversion and major-index statistics associated with $b4$, and let $c3bb$ be the Schur function. The Hall–Littlewood Schur-expansion conjecture. One has

$$\widetilde H_{\delta}(X;q,t)=\sum_{\lambda\vdash n}\sum_{\pi\in\operatorname{SYam}(\lambda)}q^{\operatorname{inv}_{\delta}(f_{\delta}(\pi))}t^{\operatorname{maj}_{\delta}(f_{\delta}(\pi))}s_{\lambda}.$$

This proposes a full Schur expansion of the modified Hall–Littlewood polynomial using the same injective map that describes the Schur expansion on each dual equivalence component. Its status depends on the preceding conjecture and is not resolved in the supplied text.