Hermite functions: Global maxima form decreasing sequence?

Question

Let \begin{equation} \psi_n(x) = \left(\frac{1}{\pi} \right)^{\frac{1}{4}}\frac{1}{\sqrt{2^n n!}} H_n(x) e^{- \frac{x^2}{2}} \end{equation} be the n-th Hermite function, where $H_n(x)$ denotes the "n-th physicists' Hermite polynomial". If one considers the plots of $\psi_n$ for different values of $n$ (cp. https://en.wikipedia.org/wiki/Hermite_polynomials#/media/File:Herm50.svg ) it looks like, that the sequence $(\sup_{x \in \mathbb{R}} |\psi_n(x)|)_{n \in \mathbb{N}}$ is decreasing. Unfortunately I was not able to find any result in this direction. Do you know some results regarding this sequence?

Answer 1

This is proven in the following paper

An Inequality for Hermite Polynomials, Jack Indritz