Plessisiso's Bogomolov property conjecture for torsion extensions of abelian varieties

Let $A$ be an abelian variety defined over a number field $K$, let $\mathcal{L}$ be a symmetric ample line bundle on $A/K$, and let $L/K$ be a finite extension. For the Néron–Tate height $\hat{h}_A$ attached to $\mathcal{L}$, say that a subgroup of $A(\overline{K})$ has property $(B)$ if its non-torsion points have height bounded below by a positive constant. Write $A_{\mathrm{tors}}$ for the torsion subgroup of $A(\overline{K})$. Plessisiso's conjecture. The fields $L(A_{\mathrm{tors}})$ and the groups $A(L(A_{\mathrm{tors}}))$ have property $(B)$. This is a special case of a conjecture attributed to the author and concerns uniform positive lower bounds for canonical heights after adjoining all torsion points; the source does not provide evidence that it has been resolved.