The asymptotic equidistribution conjecture for singleton double descent sets

Let $dd(I;n)$ denote the number of permutations in the symmetric group on $n$ letters whose double descent set is $I$. For fixed $0<\alpha<\beta<1$, the asymptotic equidistribution conjecture asserts

$$\sum_{\alpha n<i<\beta n}dd(\{i\};n)\sim(\beta-\alpha)\sum_{i=2}^{n-1}dd(\{i\};n).$$

This predicts that singleton double descents become uniformly distributed across the interior positions as $n$ grows, based on computer-generated data; its asymptotic validity remains open.