The alternating ratio 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 $n,i\in\mathbb{N}$ with $i<\left\lceil\frac n2\right\rceil-1$, the alternating ratio conjecture asserts

$$\frac{dd(\{i\};n)}{dd(\{i+1\};n)}>\frac{dd(\{i+2\};n)}{dd(\{i+3\};n)}$$

when $i$ is even, and

$$\frac{dd(\{i\};n)}{dd(\{i+1\};n)}<\frac{dd(\{i+2\};n)}{dd(\{i+3\};n)}$$

when $i$ is odd. This describes alternating monotonicity of successive ratios as the counts approach uniformity; it remains open.