Lower bound for the canonical on Abelian varieties over totally $p$-adic extensions

Let $A$ be an abelian variety defined over a number field $\mathbb{Q}$, and let $\widehat{h}$ be the Néron-Tate height on $A(\bar{\mathbb{Q}})$ corresponding to a symmetric ample line bundle on $A$. In this article, we prove that the Néron-Tate height of totally $p$-adic points is bounded below by an absolute constant depending only on $A$ for all but finitely many primes. In other words, if we denote by ${\mathbb{Q}}^{(p)}$ the maximal algebraic extension of $\mathbb{Q}$ in which $p$ is totally split, then $A({\mathbb{Q}}^{(p)})$ satisfies the Bogomolov property for all but finitely many primes. In particular, if $A$ has good reduction at a prime $p$, we obtain the Bogomolov property $A({\text{Q}}^{(p)})$. This is the first instance where such a result has been obtained in the good reduction case. In a more general setting, if $A\mathrm{/}K$ is an abelian variety and $\mathcal{K}\mathrm{/}K$ is an asymptotically positive extension as defined in \cite{AB-SK}, which includes infinite Galois extensions with finite local degree at a non-archimedean place, then $A(\mathcal{K})$ satisfies the Bogomolov property.