Character-sum formula for primary characters of unitary groups

Let $\chi=\chi^{\phi,\lambda}$ be a primary irreducible character of $\mathrm{GU}_n(q)$, where $\phi$ is an orbit other than $\{\mathbf{1}\}$ and $\mathbf{1}$ is the trivial character of $\mathrm{GU}_1(q)$. Character-sum conjecture. For every relevant integer $k$,

$$\sum_{g:\,\dim\operatorname{fix}(g)=k}\frac{\chi(g)}{\chi(1)}=(-1)^k(-q)^{\binom{k}{2}}\left[\begin{matrix}n\k\end{matrix}\right]_{-q}.$$

The conjecture predicts that the normalized character sums over elements with fixed-space dimension $k$ have the same simple form in the unitary case as in analogous general-linear computations; the paper presents this as an experimentally suggested extension.