The linear recurrence conjecture for Jack-polynomial coefficients

Let $\alpha$ be fixed, let $\rho$ be a partition, and let $a_{\rho}^{k,(\alpha)}$ be the coefficients defined by

$$h_k(A_\lambda^{(\alpha)})=\sum_\mu a_\mu^{k,(\alpha)}\theta_\mu^{(\alpha)}(\lambda).$$

For a partition $\rho$, write $\rho\cup(m)$ for adjoining a part $m$, $\rho\cup(r,s)$ for adjoining parts $r$ and $s$, and $\rho\backslash\rho_i\cup(\rho_i+m)$ for removing the part $\rho_i$ and adjoining the part $\rho_i+m$. Write $\ell(\rho)$ for the number of parts of $\rho$. The linear recurrence conjecture. For any $m\geq 2$,

$$a_{\rho\cup(m)}^{k,(\alpha)}=\sum_{r+s=m\atop r,s\geq 1}a_{\rho\cup(r,s)}^{k-1,(\alpha)}+\alpha\sum_{1\leq i\leq\ell(\rho)}\rho_i a_{\rho\backslash\rho_i\cup(\rho_i+m)}^{k-1,(\alpha)}+(\alpha-1)\cdot(m-1)a_{\rho\cup(m)}^{k-1,(\alpha)}.$$

This conjecture seeks a recurrence valid for the coefficients governing complete functions in Jack-polynomial and Jucys--Murphy-element settings; it specializes to relations already known at $\alpha=1$ and $\alpha=2$, while its validity for general $\alpha$ is left open in the source.