Dominance monotonicity conjecture for unipotent tensor squares

Let $n$ be a positive integer, let $\mu$ and $\mu'$ be partitions of $n$, and let $\tau$ be a partition of $n$. Write

$$U_{(\mu,\mu,\tau)}(q):=\left\langle\mathcal{U}^{\mu}\otimes\mathcal{U}^{\mu}\otimes\mathcal{U}^{\tau},1\right\rangle_{{\rm GL}_n(\mathbb{F}_q)}.$$

For partitions of $n$, let $\mu'\trianglelefteq\mu$ mean that $\mu'$ is below $\mu$ in dominance order. Dominance monotonicity conjecture. If $U_{(\mu,\mu,\tau)}(q)\neq 0$ for every partition $\tau$ of $n$, then $U_{(\mu',\mu',\tau)}(q)\neq 0$ for every partition $\tau$ of $n$ and every partition $\mu'\trianglelefteq\mu$. This is presented as a consequence of the first part of the preceding conjecture, since dominance below a partition with first part at most $\lceil n/2\rceil$ preserves that bound. It therefore remains conjectural in the paper.