Stanley's coloured factorization conjecture for symmetric groups

Let $[m]=\{1,2,\dots,m\}$, let $\mu\vdash k$, and let $\omega_\mu$ be a fixed element of the conjugacy class $C_\mu$ in $\mathfrak{S}_k$. Let $\mathfrak{S}^{(m)}_k$ be the set of permutations of $[k]$ whose cycles are coloured by $[m]$. For $(\alpha,\psi)\in\mathfrak{S}^{(m)}_k$, write $\kappa^{(m)}_i(\alpha,\psi)$ for the number of cycles of $\alpha$ coloured $i$, and set $\kappa^{(m)}(\alpha,\psi)=(\kappa^{(m)}_1(\alpha,\psi),\kappa^{(m)}_2(\alpha,\psi),\ldots)$, $\mathbf{p}^{\kappa^{(m)}(\alpha,\psi)}=\prod_i p_i^{\kappa^{(m)}_i(\alpha,\psi)}$. Define $(\alpha,\psi)\circ\omega_\mu=(\gamma,\nu)$ by $\gamma=\alpha\omega_\mu$ and, for every cycle $u=(u_1\;u_2\;\cdots\;u_t)$ of $\gamma$, $\nu(u)=\max_{1\leq i\leq t}\{\psi(C^\alpha(u_i))\}$, where $C^\alpha(u_i)$ is the cycle of $\alpha$ containing $u_i$. Stanley's coloured factorization conjecture. One has

$$(-1)^k F_\mu(\mathbf{p};\mathbf{-q})=\sum_{(\alpha,\psi)\in\mathfrak{S}^{(m)}_k}\mathbf{p}^{\kappa^{(m)}(\alpha,\psi)}\mathbf{q}^{\kappa^{(m)}((\alpha,\psi)\circ\omega_\mu)}.$$

This gives a conjectured combinatorial interpretation of the multivariate character polynomials $F_\mu$. The assertion is known when $m=1$, corresponding to factorizations without colours, but remains open for $m>1$, even when $\mu$ has one part.