Product Hermitian Jacobi eigenfunction conjecture on the simplex

Let $m$ be a positive integer, let $k$ be the number of matrix variables, let $(\kappa_1,\dots,\kappa_{k+1})$ be the parameters of the Hermitian Jacobi operator, and let $\mathcal{P}$ denote the set of partitions. The product Hermitian Jacobi polynomials are the polynomials $P_{\tau_1,\dots,\tau_k}^{(\kappa)}$ indexed by $(\tau_1,\dots,\tau_k)\in\mathcal{P}^k$. The Hermitian Jacobi operator on the simplex is $\mathcal{G}_{\kappa_1,\dots,\kappa_{k+1}}^m$. Product Hermitian Jacobi eigenfunction conjecture. The polynomials

$$\bigl(P_{\tau_1,\dots,\tau_k}^{(\kappa)}\bigr)_{\tau_1,\dots,\tau_k\in\mathcal{P}}$$

are eigenfunctions of $\mathcal{G}_{\kappa_1,\dots,\kappa_{k+1}}^m$. The conjecture identifies the product construction as a family of eigenfunctions of the matrix-valued Jacobi operator; the cases $m=1$ and $k=1$ reduce to settings in which the corresponding eigenfunction statements are known.