Veselov's simplicity conjecture for generalized Hermite polynomial zeros

Let $H_{\lambda}(z)$ be the generalized Hermite polynomial associated with a partition $\lambda$. Veselov's conjecture. For any partition $\lambda$, the zeros of $H_{\lambda}(z)$ are simple, except possibly for the zero at $z=0$. This simplicity would ensure that the exceptional zeros of the associated exceptional Hermite polynomials converge individually to the zeros of $H_{\lambda}$. The conjecture is stated as open in the source; the paper's asymptotic result applies when the zeros of $H_{\lambda}$ are simple.