The freeness conjecture for quasi-flag representation rings

Let $W$ be the Weyl group under consideration, let $T$ be its maximal torus with representation ring $R(T)$, and let $\boldsymbol{\mathscr{Q}}_m(W)$, $\boldsymbol{\mathscr{Q}}'_m(W)$, and $\mathscr{Q}_m(W)$ be the quasi-flag representation rings defined in the paper. For $k\in\mathcal{M}(W)$, regard $\boldsymbol{\mathscr{Q}}_{2k}(W)$ and $\boldsymbol{\mathscr{Q}}'_{2k+1}(W)$ as $R(T)$-modules through the natural inclusion into $R(T)\otimes\mathbb{Z}[W]$.

Freeness conjecture. For all $k\in\mathcal{M}(W)$, after possibly inverting $|W|$, the following hold:

1. $\boldsymbol{\mathscr{Q}}_{2k}(W)$ is a free $R(T)$-module of rank $|W|$;
2. $\mathscr{Q}_{k}(W)$ is a free $R(T)^W$-module of rank $|W|$;
3. $\boldsymbol{\mathscr{Q}}'_{2k+1}(W)$ is a free $R(T)$-module of rank $|W|$.

This conjecture generalizes the freeness theorem established earlier in the paper for the corresponding non-exponential quasi-flag objects. The statement concerns the module structure and freeness of the exponential analogues; its resolution is not given in the source.