Setup. Let $p > 2$ be a prime, let $K$ be the completion of the maximal unramified extension of $\mathbb{Q}_p$, and fix an algebraic closure $\overline{K}$ of $K$. Let $A/K$ be an abelian variety of dimension $n$ with good reduction, let $\mathcal{A}/\mathcal{O}_K$ denote the Neron model of $A$, and consider the $n$-dimensional formal group $\widehat{\mathcal{A}}/\mathcal{O}_K$ associated to $\mathcal{A}$.
Suppose $P = (x_1,...,x_n)$ is a non-zero, $p$-torsion point of $\widehat{\mathcal{A}}(\mathcal{O}_{\overline{K}})$. I am interested in understanding the ramification behavior of the field $K(P)/K$. In particular, here is my question.
Question. Is the extension $K(P)/K$ tamely ramified?
I know the answer to the question is yes when $A$ is an elliptic curve $E$. Indeed, if $E$ has good ordinary reduction, then the formal group is isomorphic to $\widehat{\mathbb{G}_m}$ (recall that $\mathcal{O}_K \cong W(\overline{\mathbb{F}_p})$), in which case the result is clear, and if $E$ has good supersingular reduction, then this is true by work of Serre [1, Section 1]. One can also see that the answer is yes whenever $A$ has good ordinary reduction because the formal group is toroidal. There is also work of Arias-de-Reyna [2, Theorem 3.3] which gives a criterion saying when wild inertia will act trivially on the $p$-torsion points, however, this condition is strong, and it will not hold in general. I have searched extensively through Hazewinkel's Formal Groups and Applications, but I have not been able to find a result of this form.
Any comments, suggestions, references, and/or counterexamples would be greatly apprecaited.
References.
[1] J.-P. Serre, Propriètès galoisiennes des points d’ordre fini des courbes elliptiques
[2] S. Arias-de-Reyna, Formal groups, supersingular abelian varieties and tame ramification
