An upper-bound conjecture for weighted Hermite polynomials

Let $H_n(x)$ denote the $n$th Hermite polynomial, let

$$g_n(x)=\left(1-\frac{x^2}{2n+1}\right)^{-1/2},$$

and set

$$x_n=\sqrt{2n+1}\left(1-\frac{\pi}{2(2n+1)}\right).$$

Hermite-polynomial upper-bound conjecture. For all $n\in\mathbb{N}^+$ and $x>0$,

$$\left|H_n(x)\mathrm{e}^{-x^2}\right|\leq\frac{2^n\Gamma\left(\frac{n+1}{2}\right)}{\sqrt{\pi}}g_n(\min(x,x_n))\mathrm{e}^{-x^2/2}.$$

The conjecture is motivated by the interpretation of $H_n(x)\mathrm{e}^{-x^2/2}$ as the unnormalized wave function of a quantum oscillator and is intended to improve estimates for the remainder term in the modified Euler–Maclaurin formula, particularly for large $m$. Its resolution is not specified in the source.