Real and imaginary zero conjecture for Wronskians of doubled Hermite partitions

Let $\nu=(\mu_1^2,\dots,\mu_n^2)$ be a doubled partition with distinct parts, where $\mu_i^2$ denotes that the part $\mu_i$ occurs twice, and let $W_{\nu}(z)$ be its associated Wronskian of Hermite polynomials. Doubled-partition zero conjecture. The Wronskian $W_{\nu}(z)$ has no real roots and has as many pure imaginary roots as there are odd numbers among $\mu_1,\dots,\mu_n$. This conjecture seeks an effective description of the real zeros of $W_{\lambda}(z)$ in terms of the corresponding Young diagram. It concerns the special class of doubled partitions and complements the preceding simplicity conjecture by specifying the location and count of roots in this case.