The sharp Hermite-ratio upper-bound conjecture

Let $h_n(x)$ and $F_n(x)=h_n(x)/h_{n+1}(x)$ be the ratios considered above, and let $g_{\alpha,\beta}(x)$ denote the comparison function introduced in the paper. Hermite-ratio upper-bound conjecture.

$$F_n(x)<g_{-\frac12,\frac12}(x)$$

for all real $x$ and $n>\frac12$. This conjecture proposes the comparison function with parameters $\alpha=-\frac12$ and $\beta=\frac12$, which has the exact limiting values of $F_n(x)$ as $x\to\pm\infty$; its status is not resolved in the supplied text.