Strong triangularity conjecture for the Jacobi operator on the Hermitian matrix simplex

Let $\mathcal{G}_{\kappa_1,\dots,\kappa_{k+1}}^m$ be the Hermitian Jacobi operator on the simplex, and let the multivariate Schur polynomials be the basis of unitary-conjugation-invariant polynomials used to organize polynomial degrees. Strong triangularity conjecture. The operator $\mathcal{G}_{\kappa_1,\dots,\kappa_{k+1}}^m$ is strongly triangular with respect to the multivariate Schur functions. Partial results are given in this direction, and the conjecture is intended to describe the operator's action in the multivariate Schur basis and thereby support the construction of eigenfunctions.