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 $\hat{h}$ be the N\'eron-Tate height on $A(\overline{\mathbb{Q}})$ corresponding to a symmetric ample line bundle on $A$. In this article, we prove that the N\'eron-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 $A(\Q^{(p)})$. This is the first instance where such a result has been obtained in the good reduction case. In a more general setting, if $A/K$ is an abelian variety and $\mathcal{K}/K$ 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.