Conjecture of simple zeros for generalized Jacobi polynomials

Let $\lambda$ and $\mu$ be partitions, and let $m_1$ denote the relevant parameter associated with the partitions. Take parameters $\alpha>-1$ and $\beta>m_1$ satisfying the conditions required for the generalized Jacobi polynomial. If $\lambda$ is an even partition, then the generalized Jacobi polynomial $\Omega^{(\alpha,\beta)}_{\lambda,\mu}$ has only simple zeros.

Simple-zero conjecture. For any partitions $\lambda$ and $\mu$, take $\alpha>-1$ and $\beta>m_1$ such that the stated conditions are satisfied. If $\lambda$ is an even partition, then the zeros of $\Omega^{(\alpha,\beta)}_{\lambda,\mu}$ are simple.

This conjecture proposes that the simple-zero assumption used in the theorem on exceptional zeros is not a restriction for the relevant generalized Jacobi polynomials. The surrounding discussion notes that nonsimple zeros found computationally occur when the corresponding exceptional Jacobi polynomials do not form a complete set; the conjecture itself is presented without a resolution.