Simplicity conjecture for nonzero roots of generalized Wronskian–Hermite polynomials

Let $W_N^{[m,k,l]}(z)$ denote the generalized Wronskian–Hermite polynomial associated with the arithmetic-progression parameters $N,m,k,l$. The parameters satisfy

$$N\ge1,\qquad m\ge2,\qquad k\ge2,\qquad 1\le l\le k-1.$$

Simplicity conjecture. All nonzero roots of $W_N^{[m,k,l]}(z)$ are simple.

The problem arose from the study of rogue-wave patterns for integrable equations. The conjecture is proved in the paper for three subclasses of these polynomials, covering more than half of the relevant parameter range, but remains open in general.