The alternating-sum conjecture for zeros of Wronskians of orthogonal polynomials

Let $\{P_n\}_{n=0}^\infty$ be an orthogonal polynomial system with respect to an arbitrary positive measure $d\mu=W\,dx$ supported on an interval $I$ of the real line. Assume that $\{P_n\}_{n=0}^\infty$ is non-degenerate in the sense of Definition 5.1, and let $P_\lambda$ denote the Wronskian determinant associated with an arbitrary sequence indexed by a partition $\lambda=(\lambda_1,\ldots,\lambda_\ell)$. Alternating-sum conjecture. The Wronskian determinant $P_\lambda$ has

$$n(P_\lambda)=\sum_{j=1}^\ell (-1)^{\ell-j}\lambda_j$$

simple real zeros in the support of $\mu$. This would extend the corresponding zero-counting result beyond eigenfunction systems to arbitrary orthogonal polynomial systems, generalizing a conjecture of Durán.