Strengthened West conjecture for average stack-sorting depth

Let $\mathcal D_n$ be the average number of iterations of the stack-sorting map needed to sort a permutation in $S_n$ into the identity permutation. Strengthened West conjecture. The limit

$$\lim_{n\to\infty}\frac{\mathcal D_n}{n}$$

exists and lies in the interval $(0.77,0.81)$. This strengthens West's conjecture by proposing numerical bounds in addition to existence; the source gives numerical evidence but no proof.