Veselov's conjecture on simple zeros of Wronskians of Hermite polynomials

Let $n_1,n_2,\ldots,n_r$ be positive integers, and let $H_{n_i}(x)$ denote the Hermite polynomials. Their Wronskian is

$$\operatorname{Wr}[H_{n_1}(x),H_{n_2}(x),\ldots,H_{n_r}(x)].$$

Veselov's conjecture. This Wronskian has simple zeros, except possibly at $x=0$.

The conjecture concerns the multiplicities of zeros of Wronskians of Hermite polynomials and is connected to the irreducibility of the associated remainder polynomials. It is known in a few cases, but remains open in general.