Bogomolov property of some infinite nonabelian extensions of a totally $v$-adic field

Let $E$ be an elliptic curve defined over a number field $K$, and let $v$ be a finite place of $K$. Write $K^{tv}$ for the maximal totally $v$-adic field, and denote by $L$ the field generated over $K^{tv}$ by all torsion points of $E$. Under some conditions, we will show that the absolute logarithmic Weil height (resp. Néron–Tate height) of any element of $L$ (resp. $E(L)$) is either $0$ or bounded from below by a positive constant depending only on $E,K$ and $v$. This constant will be explicit in the toric case.