Felder–Hemery–Veselov conjecture on zeros of Wronskians of Hermite polynomials

Let $H_\lambda$ denote the Wronskian Hermite polynomial associated with a partition $\lambda$. A doubled partition has the form

$$\lambda=(\mu_1^2,\dots,\mu_n^2).$$

Felder–Hemery–Veselov conjecture. For every doubled partition $\lambda=(\mu_1^2,\dots,\mu_n^2)$, $H_\lambda$ has no real roots and has as many imaginary roots as there are odd numbers in the partition.

This conjecture concerns the location and multiplicity-free behavior of roots of Wronskians of Hermite polynomials. The source attributes it to Felder et al.; it is stated there as a conjecture, and no resolution is supplied.