Murty–Patankar's conjecture on simple reductions of abelian varieties

Let $A$ be an absolutely simple abelian variety over a number field $K$. Let $\mathcal V$ be the set of finite places $v$ of $K$ for which $A$ has good reduction and $A_v/\mathbb F_v$ is simple. The density of $\mathcal V$ should exist after possibly replacing $K$ by a finite extension, and it should equal $1$ if and only if $\operatorname{End}(A_{\overline K})$ is commutative. This conjecture predicts that the splitting behavior of reductions is governed by the geometric endomorphism ring; the paper supplies evidence and proves many cases, while the general assertion remains open.