Murty–Patankar splitting conjecture for absolutely simple abelian varieties

Let $X$ be an absolutely simple abelian variety over a number field $K$. Say that a prime of $K$ is a splitting prime for $X$ when the reduction of $X$ at that prime is not simple. Murty–Patankar's splitting conjecture. The set of primes of $K$ where $X$ splits has positive density if and only if $\operatorname{End}_{\bar K}(X)$ is noncommutative.

This conjecture predicts that positive-density splitting of an absolutely simple abelian variety is governed precisely by whether its geometric endomorphism ring is noncommutative. The supplied text gives no resolution, so its status remains open.