  Let $k$ be a number field and let ${\mathcal{A}}$ be a ${\rm GL}_2$-type
variety defined over $k$ of dimension $d$. We show that for every prime number
$p$ satisfying certain conditions (see Theorem 2), if the local-global
divisibility principle by a power of $p$ does not hold for ${\mathcal{A}}$ over
$k$, then there exists a cyclic extension $\widetilde{k}$ of $k$ of degree
bounded by a constant depending on $d$ such that ${\mathcal{A}}$ is
$\widetilde{k}$-isogenous to a ${\rm GL}_2$-type variety defined over
$\widetilde{k}$ that admits a $\widetilde{k}$-rational point of order $p$.
Moreover, we explain how our result is related to a question of Cassels on the
divisibility of the Tate-Shafarevich group, studied by Ciperiani and Stix and
Creutz.
