Strict log-concavity conjecture for the rank of strongly unimodal sequences

Let $u(m,n)$ denote the number of strongly unimodal sequences of size $n$ and rank $m$. Strict log-concavity conjecture. For $n\ge37$ and $|m|\le n-23$, we have

$$u(m,n)^2 > u(m-1,n)u(m+1,n).$$

The conjecture is known asymptotically: for each fixed $m$, the inequality holds as $n$ tends to infinity. The stated uniform range remains conjectural.