Asymptotic average-sorting-time conjecture for type-B stack-sorting

Let $\mathcal D_n$ be the average number of iterations of the ordinary stack-sorting map $\operatorname{\mathtt{s}}$ needed to send an element of $S_n$ to the identity permutation, and let $\mathcal D_n^B$ be the average number of iterations of the type-$B$ stack-sorting map $\operatorname{\mathtt{s}}_B$ needed to send an element of $B_n$ to the identity element. Asymptotic average-sorting-time conjecture. One has

$$\lim_{n\to\infty}\left(\frac{\mathcal D_n}{n}-\frac{\mathcal D_n^B}{n}\right)=0.$$

The conjecture asserts that the normalized average numbers of iterations agree asymptotically between types $A$ and $B$. The source gives no resolution, so the conjecture remains open.