This question is related to: https://math.stackexchange.com/q/4270522/168758

Let $H_n$ be the probabilist's $n$th Hermite polynomial. These are given explicitly by the following formula (which ever halps)

$$ \begin{split} H_n(x) &= n!\sum_{k=0}^{\lfloor n/2\rfloor}\frac{(-1)^k}{2^kk!(n-2k)!}x^{n-2k}\\ H_n(x) &= \frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty (x+iy)^n e^{-y^2/2}dy\\ H_n(x) &= e^{-D^2/2}x^n, \end{split} $$ where $D^2$ is the second-derivative-w.r.t-$x$ differential operator $\dfrac{d^2}{dx^2}$, and $e^{-D^2/2}$ should be seen as a power series in $D^2$.

Let $d$ be a large positive integer, $a$ and $b$ be fixed vectors on the unit $(d-1)$-dimensional sphere $S_{d-1}$, and $X$ be uniformly distributed on $S_{d-1}$. For fixed nonnegative integers $n$ and $m$, lefine

$$ s_{n,m} = \mathbb E[H_n(X^\top a)H_m(X^\top b)]. $$ Due to rotational-invarfiance of $X$, it is clear that $s_{n,m}$ is a polynomial in $t:=a^\top b$. Let $c_{n,m,k}$ be the coefficient of $t^k$ in $s$.

Question. For $k \ge 1$, what is a good Big-O upper-bound for $c_{n,m,k}$ in the limit $d \to \infty$.