The symmetric-polynomial form conjecture for transitive factorisation generating series

Let $u,z,p_1,p_2,\ldots$ be indeterminates, let $p_\alpha=p_{\alpha_1}p_{\alpha_2}\cdots$, and define

$$F_k^{(m)}(u,z;p_1,p_2,\ldots)=\sum_{\substack{n\geq1\\\\ k-1\mid n+m-2}}\sum_{\substack{\alpha\vdash n\\\\ l(\alpha)=m}}c_k(\alpha)\\,\lvert\mathcal{C}_\alpha\rvert\\,p_\alpha\frac{u^{\mu_k(\alpha)}}{\mu_k(\alpha)!}\frac{z^n}{n!}.$$

For a partition $\alpha$ with $m$ parts, define $\psi_m(p_\alpha u^i z^j)=\sum_{\sigma\in\mathfrak{S}_m}x_1^{\alpha_{\sigma(1)}}\cdots x_m^{\alpha_{\sigma(m)}}$, and set $P_k^{(m)}(x_1,\ldots,x_m)=\psi_m(F_k^{(m)})$. Let $w_i=w(x_i)$, where $w(x)$ is the unique power-series solution of the functional equation given in the source. The symmetric-polynomial form conjecture. For $m\geq1$,

$$\left(\sum_{i=1}^m x_i\frac{\partial}{\partial x_i}\right)^{3-m}P_k^{(m)}(x_1,\ldots,x_m)=S_k^{(m)}(w_1,\ldots,w_m)\prod_{i=1}^m x_i\frac{d w_i}{d x_i},$$

where $S_k^{(m)}(w_1,\ldots,w_m)$ is a symmetric polynomial in $w_1,\ldots,w_m$. The conjecture gives a uniform algebraic form for the generating series of minimal transitive ordered factorisations; the later discussion conjectures polynomial dependence on $k$ for the coefficients of the symmetric polynomial, but does not establish the general case.