The down-up-down-up 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 $I$. Given a fixed $n\in\mathbb{N}$, the down-up-down-up conjecture asserts that for $2\leq i<\left\lceil\frac n2\right\rceil$, $dd(\{i\};n)>dd(\{i+1\};n)$ when $i$ is even, and $dd(\{i\};n)<dd(\{i+1\};n)$ when $i$ is odd. The pattern has been numerically verified for some values of $n$ and is conjectured to persist generally.