Garsia–Procesi modules are weakly quasisymmetric characteristic compatible

Let $\falR_{\fmu}$ denote the Garsia–Procesi module associated with a partition $\fmu$, and let $\fY={x^{\alpha}\mid \alpha\text{ is sub-Yamanouchi}}$ be its specified monomial basis, ordered by $\fle_r$. A triple $(M,\fB,\fle)$ is called weakly quasisymmetric characteristic compatible (WQCC) when the weak Hecke deformation $\foverline{\pi}_i$ of the $\fS_n$-action defines an $\fH_n(0)$-module $\fwidehat M$ satisfying $F_{char}^Q(\fwidehat M)=F_{char}(M)$. Garsia–Procesi WQCC conjecture. If \fY=\{x^{\alpha}\mid \alpha\text{ is sub-Yamanouchi}}, then $(\fR_{\fmu},\fY,\fle_r)$ is WQCC. Extensive computer simulation suggests this additional WQCC example beyond the proved coinvariant example, but no proof or resolution is given here.