Triviality of torsors after a field extension of bounded degree

Question

Let $G$ be an abelian variety defined over a ring $R$. Is there a natural number $n$ such that, for any field $k$ over $R$ and any $G_k$-torsor $T$, there exists an extension $L/k$ of degree $n$ for which $T(L)$ is not empty?

Answer 1

This is not even true for elliptic curves over $\mathbb Q$ or $\mathbb Q_p$. For example, by Tate duality the group $H^1(\mathbb Q_p,E)$ is dual to $E(\mathbb Q_p)$, which is an infinite group. This shows that the period can be arbitrarily large, and then one can use the fact that the period divides the index, to show that the index can be arbitrarily large.