Conjecture on common non-zero zeros of Hermite polynomials

Let $\mathit{He}_n(x)$ denote the probabilists' Hermite polynomial of degree $n$, defined by

$$\mathit{He}_n(x)=(-1)^n\exp(x^2/2)\frac{\mathrm{d}^n}{\mathrm{d}x^n}\exp(-x^2/2).$$

The preceding properties imply that consecutive Hermite polynomials have no common zeros. Hermite zero-separation conjecture. For any $m,n\in\mathbb{N}$ with $m\neq n$, there is no common non-zero zero of $\mathit{He}_n(x)$ and $\mathit{He}_m(x)$; equivalently,

$$\nexists x\in\mathbb{R}\backslash\{0\}\text{ such that }\mathit{He}_n(x)=\mathit{He}_m(x)=0.$$

The conjecture concerns the distribution of zeros of Hermite polynomials beyond the known interlacing and coprimality results for consecutive degrees. The source states that no proof was known to the authors.