Central limit conjecture for descents after stack-sorting

For each $n\geq 1$, let $D_n$ be the number of descents of the stack-sorted image of a uniformly random permutation in $S_n$, and define

$$Y_n=\frac{D_n-(3-e)n}{\sqrt{n}}.$$

Central limit conjecture. The sequence $(Y_n)_{n\geq 1}$ converges in distribution to a random variable $Y$ such that

$$Y\sim N(0,2+2e-e^2).$$

The conjecture is motivated by the computed asymptotics of the first six central moments, which agree with those of an asymptotically normal distribution; its resolution is not given in the source.