Defant's linear-time conjecture for Pop-Stack Sorting on random permutations

Let $D70E_n$ be a uniformly random permutation in the symmetric group $D70E_n$, and let $D43E\operatorname{Pop}$ denote one step of Pop-Stack Sorting, which reverses every maximal decreasing run of a permutation. Write $D43E\operatorname{Id}_n$ for the identity permutation and let $o(1)$ tend to zero as $n\to\infty$. Defant's conjecture. Almost surely, Pop-Stack Sorting needs

$$(1-o(1))n$$

steps to reach $D43E\operatorname{Id}_n$ from $D70E_n$. This conjecture predicts a linear lower bound, asymptotically matching the trivial upper bound up to a vanishing factor; its resolution concerns the typical running time of Pop-Stack Sorting on a uniformly random permutation.