The positive ratio-limit conjecture for singleton double descent counts

Let $dd(I;n)$ denote the number of permutations in the symmetric group on $n$ letters whose double descent set is $I$. For fixed $i,j\in\mathbb{Z}_{\geq2}$, the positive ratio-limit conjecture asserts that

$$\lim_{n\to\infty}\frac{dd(\{i\};n)}{dd(\{j\};n)}$$

exists and is positive. This predicts a sharp contrast with ordinary descent counts, whose corresponding ratios are generally zero or infinite; the conjecture remains open.