The colored-permutation formula conjecture for normalized character polynomials

Let $\lambda$ be the partition of $n$ whose diagram is the union of $m$ rectangles described above. Let $\mu\vdash k$ and fix a permutation $w_\mu\in\mathfrak S_k$ of cycle type $\mu$. Let $\mathfrak S_k^{(m)}$ be the set of permutations in $\mathfrak S_k$ whose cycles are colored by $[m]=\{1,\dots,m\}$. For $\alpha=(u,\varphi)$ and $v\in\mathfrak S_k$, define $\alpha v=(uv,\psi)$ by assigning to each cycle of $uv$ the maximum color among the cycles of $u$ contributing to it. If $\kappa_i(\alpha)$ is the number of cycles of $u$ colored $i$, write $\boldsymbol p^{\kappa(\alpha)}=\prod_i p_i^{\kappa_i(\alpha)}$, and similarly for $\boldsymbol q$. Colored-permutation formula conjecture. One has

$$F_\mu(\boldsymbol p;\boldsymbol q)=(-1)^k\sum_{\alpha w_\mu=\beta}\boldsymbol p^{\kappa(\alpha)}(-\boldsymbol q)^{\kappa(\beta)},$$

where the sum ranges over all $(k+m-1)_k$ pairs $(\alpha,\beta)\in\mathfrak S_k^{(m)}\times\mathfrak S_k^{(m)}$ satisfying $\alpha w_\mu=\beta$. This conjecture generalizes the preceding permutation formula, which is stated as a theorem in the source; its status is not resolved in the supplied text.