Intermediate Macdonald polynomial conjecture for transitive Weyl actions

Let $(\mathbf {U},\mathbf{B},\gamma)$ be a commutative triple, and suppose that the action of $W_\Sigma$ on $\mathfrak{B}(\gamma)$ is transitive and that one stabiliser $W_J$ is a parabolic subgroup of $W_\Sigma$. Let MSF denote the matrix spherical functions and let their restrictions be shifted by $-\rho$. Intermediate Macdonald polynomial conjecture. The $-\rho$-shifted restrictions of the MSF for $(\mathbf {U},\mathbf{B},\gamma)$ are Intermediate Macdonald polynomials invariant under $W_J$. This is the proposed generalisation from the examples considered in the paper; the supplied text gives no evidence resolving it.