S. David's Bogomolov property conjecture for torsion fields of abelian varieties

Let $A$ be an abelian variety defined over a number field, and let $A_{\mathrm{tors}}$ denote its torsion subgroup. S. David's conjecture. The field $\mathbb{Q}(A_{\mathrm{tors}})$ has the Bogomolov property. This extends Habegger's theorem for torsion fields of elliptic curves; the supplied status evidence indicates that the surrounding modularity result is resolved, but does not resolve this conjecture itself.