Conjectured limiting average depth for pop-stack-sorting

Let $\mathsf{Pop}$ be the pop-stack-sorting map, which reverses every maximal decreasing run of a permutation, and let $\mathcal D_n^{\mathsf{Pop}}$ be the average number of iterations of $\mathsf{Pop}$ needed to sort a permutation in $S_n$ into the identity permutation. Pop-stack-sorting average-depth conjecture.

$$\lim_{n\to\infty}\frac{\mathcal D_n^{\mathsf{Pop}}}{n}=1.$$

The source states that little is known about the average number of pop-stack-sorting iterations and offers this limit as a conjecture for further study.