Bogomolov property conjecture from bounded local degree

Let $K$ be a number field, let $v$ be a finite place of $K$, and let $L/K$ be an algebraic extension. For an extension $w$ of $v$ to $L$, write $L_w$ for the corresponding completion and define the local degree

$$d_v(L)=\sup_w [L_w:K_v].$$

Let $A$ be an abelian variety over $K$, let $\mathcal{L}$ be a symmetric ample line bundle on $A/K$, and let $A_{\mathrm{tors}}$ denote its torsion subgroup. Bounded-local-degree conjecture. If $d_v(L)$ is finite for at least one finite place $v$ of $K$, then $L(A_{\mathrm{tors}})$ and $A(L(A_{\mathrm{tors}}))$ have property $(B)$, meaning that the relevant non-torsion points have positive canonical height bounded away from zero. This conjecture would encompass the two preceding theorems in the paper and proposes bounded local degree at a single finite place as a sufficient condition for the Bogomolov property; the source gives no evidence of a resolution.