Simple-zero conjecture for Wronskians of Hermite polynomials

Let $\lambda$ be a partition, and let $W_{\lambda}(z)$ denote the Wronskian of Hermite polynomials associated with $\lambda$. Simple-zero conjecture. For every partition $\lambda$, all the zeroes of $W_{\lambda}(z)$ are simple except possibly for $z=0$. This property was conjectured in connection with the locus problem for the zeros of Wronskians of Hermite polynomials; it would provide a more effective proof of Oblomkov's result. The possible multiplicity at $z=0$ can be computed explicitly, and triangular Young diagrams give the extreme case in which all zeros occur at the origin.