Thompson's $q$-power degree conjecture for unitriangular groups

Let $\mathbb{F}_q$ be a finite field, let $n\geq 1$, and let $U_n(\mathbb{F}_q)$ be the group of unitriangular $n\times n$ matrices over $\mathbb{F}_q$. Thompson's $q$-power degree conjecture. Every irreducible complex representation of $U_n(\mathbb{F}_q)$ has degree equal to a power of $q$. The claim concerns the degree pattern of irreducible representations of unitriangular groups, which are finite $p$-groups; no resolution evidence is supplied in the text.