Pairwise orthogonality conjecture for multivariate Hermitian Jacobi polynomials

Let $\Theta_k$ index the multivariate Hermitian Jacobi polynomials $\tilde{P}^{(\alpha,\beta)}_{\tau_1,\dots,\tau_k;\phi}$, defined by triangular expansion in the multivariate Schur polynomials with respect to the matrix beta distribution $MB_{k,m}^{(\alpha,\beta)}$. Let $(\tau_1,\dots,\tau_k;\phi)$ and $(\rho_1,\dots,\rho_k;\psi)$ be non-equivalent indices. Pairwise orthogonality conjecture. The polynomials $\tilde{P}^{(\alpha,\beta)}_{\tau_1,\dots,\tau_k;\phi}$ and $\tilde{P}^{(\alpha,\beta)}_{\rho_1,\dots,\rho_k;\psi}$ are pairwise orthogonal for non-equivalent indices. This extends the evident orthogonality for comparable indices to all non-equivalent indices; the source notes that orthogonality for equal degree data can depend on the choice of multivariate Schur polynomials.