Unimodality conjecture for major index on two-row standard Young tableaux

Let $f_{(n-k,k),i}(q)$ be the generating polynomial for the major index of standard Young tableaux of shape $(n-k,k)$ having $i$ descents. Thus the relevant parameters are integers $n$, $k$, and $i$, in the range in which such tableaux exist.

Two-row tableau unimodality conjecture. For all $n$, $k$, and $i$, the polynomial $f_{(n-k,k),i}(q)$ is symmetric and unimodal, with central term $ni/2$.

The paper says this statement was empirically verified for $n\leq 30$ and that it would imply the major-index unimodality conjecture above through the displayed decomposition of $A_{n,i}(q)$. No proof of the statement is given, so it remains open in the source.