Eigenvalue-refined factorization conjecture for unitary groups

Let $\mathrm{GL}^\varepsilon_n$ denote the general linear group when $\varepsilon=1$ and the general unitary group when $\varepsilon=-1$, and let $\mathrm{GL}^\varepsilon_1$ be its one-dimensional analogue. Let $c$ be an irreducible element of $\mathrm{GL}^\varepsilon_n$, and let $(a_1,\ldots,a_\ell)$ be elements of $\mathrm{GL}^\varepsilon_1$ satisfying $a_1\cdots a_\ell=c$. Suppose exactly $m$ of the $a_i$ equal $1$. Eigenvalue-refined factorization conjecture. The number of factorizations $c=t_1\cdots t_\ell$ into reflections with $\det(t_i)=a_i$ for every $i$ is

$$[n]_{\varepsilon q}^{\ell-1}\sum_{i=0}^{\min(m,\ell-n)}(-1)^i\binom{m}{i}\left[\begin{matrix}\ell-i-1\ -1\end{matrix}\right]_{\varepsilon q}.$$

This conjecture proposes that the eigenvalue-refined enumeration known for general linear groups extends to the unitary case; it would, together with the stated counting fact for products of nonidentity elements, imply the corresponding unrefined factorization formula.