The strong-torsion bound for regular realizations

Let $G$ be a finite group with a regular realization over $\mathbb Q$, let $r$ be the number of branch points, let $m$ be the index of a relevant Sylow subgroup, and let $p^u$ be the order of the abelianization of a $p$-Sylow subgroup.

Strong-torsion bound conjecture. Some expression in $r$ and $m$, independent of the prime of good reduction $v$, bounds $p^u$.

The source says this follows from the Strong Torsion Conjecture on abelian varieties and that it would yield forms of the Main modular-tower conjecture.