The bounded--power conjecture for regular realizations

Let $G$ be a finite group with a regular realization over $\mathbb Q$, let $P_\ell$ be an $\ell$-Sylow subgroup of $G$, let $m=[G:P_\ell]$, and let $r$ be the branch-point number of the realizing cover. If the abelianization of $P_\ell$ has order $\ell^u$, then $u$ measures the relevant $\ell$-power torsion size. Bounded-$\ell$-power conjecture. Some expression in $r$ and $m$, independent of the good-reduction prime $\nu$, bounds $\ell^u$. The claim is presented as a stronger form of the main regular inverse Galois problem conjecture and as following from the torsion conjecture for abelian varieties; the source does not establish it.