Dense globally F-split type conjecture for varieties with int-amplified endomorphisms

Let $X$ be a normal projective variety over an algebraically closed field $k$ of characteristic zero. An endomorphism $f\colon X\to X$ is int-amplified if there exists an ample Cartier divisor $H$ on $X$ such that $f^*H-H$ is ample. The variety $X$ is of dense globally $F$-split type if its reduction modulo $p$ is globally $F$-split for infinitely many primes $p$. Dense globally $F$-split type conjecture. If $X$ admits an int-amplified endomorphism, then $X$ is of dense globally $F$-split type. An affirmative answer to Schwede and Smith's question whether varieties of dense globally $F$-split type are of Calabi–Yau type would imply the preceding Calabi–Yau type conjecture; this statement is open in the general setting.