The top Gallai-polynomial conjecture via second-order Eulerian numbers

For $n>1$, let $Q_{n,n-1}(q)$ be the coefficient appearing in the conjectured Hilbert-series expansion of the Gallai algebra $\mathcal G_{n,k}$. Let $E(n-1,j)$ denote the second-order Eulerian number, counting Stirling permutations of order $n-1$ with $j$ descents. Top Gallai-polynomial conjecture. For every $n>1$,

$$Q_{n,n-1}(q)=q^{\binom{n}{2}-1}\sum_{j=0}^{n-1}E(n-1,j)q^{-j}.$$

This gives an explicit formula for the highest-index coefficient in the Gallai Hilbert-series expansion; the source provides no resolution beyond the conjectural statement.