Uniform factorization-count hypothesis for regular elliptic elements

Let $g$ be a regular elliptic element of $GL_n({\mathbb{F}}_q)$, and let $t_q(n,\ell)$ denote the number of ordered reflection factorizations of a Singer cycle of length $\ell$. Uniform factorization-count hypothesis. The number of ordered reflection factorizations $g=t_1 t_2 \cdots t_\ell$ is the same for all regular elliptic elements $g$ in $GL_n({\mathbb{F}}_q)$, namely $t_q(n,\ell)$. This is supported by empirical evidence and would extend the factorization enumeration from Singer cycles to all regular elliptic elements; no resolution is supplied in the source.