The conjecture on real periodic points of Hermite polynomials

For each $d\geq 1$, define the Hermite polynomial by

$$H_d(z)=(-1)^d e^{z^2}\frac{\mathrm{d}^d}{\mathrm{d}z^d}e^{-z^2}.$$

Let $\mathscr{R}_d$ denote the set of real polynomials of degree $d$ having only real periodic points. Hermite polynomial conjecture. For any $d\geq 1$, the Hermite polynomial $H_d\in \mathbb{R}[x]_d$ has only real periodic points, that is, $H_d\in \mathscr{R}_d$. The conjecture is motivated by the verified cases $d=3$ and $d=4$ and by computer experiments; its general status is not resolved in the source.