Let $H_k(x)$ be (probabilists' or physicists', does not matter for this question) Hermite polynomials.

It is well-known that all the gaps between consecutive roots of $H_k(x)$ are at least a multiple of $1 / \sqrt{k}$ (see, e.g., Szego, "Orthogonal Polynomials", page 130).

Is the same known to be true if we consider the roots of $H_k(x)$ and $H_{k+1}(x)$ together? Equivalently, is it true that the zeros of $H_{k+1}(x)$ are roughly half-way between its respective local minima and maxima?

Looking at the numerical values for $k \leq 100$, the desired bound seems to be true.