The growth constant for the stack-sorting map

Let $S_n$ be the symmetric group, let $s:S_n\to S_n$ be the stack-sorting map, and let $\deg(s:S_n\to S_n)$ denote its maximum fiber size. The paper considers the exponential growth rate

$$\lim\limits_{n\to\infty}\deg(s:S_n\to S_n)^{1/n}.$$

Stack-sorting growth conjecture. The value of this limit lies in the interval $(1.68,1.73)$.

The interval is suggested by computations on random permutations for $n=100$ and $n=300$. The problem of obtaining improved asymptotic estimates, or exact formulas, remains open.