Quasi-$F$-splitting of general fibers with trivial relative canonical divisor

Let $\pi \colon X \to Y$ be a fiber space of normal varieties. The relative canonical divisor is trivial if $K_X-\pi^*K_Y$ is linearly equivalent to zero, and $X$ is quasi-$F$-split when it satisfies the quasi-$F$-splitting condition defined in the paper.

General-fiber quasi-$F$-splitting conjecture. If the relative canonical divisor is trivial and $X$ is quasi-$F$-split, then a general fiber of $\pi$ is quasi-$F$-split.

The assumption that the generic fiber is a complete intersection in the preceding corollary is technical, motivating this conjecture. It is disproved in general when the relative canonical divisor is anti-ample, as shown by the paper's wild conic bundle example; the stated trivial-divisor case is therefore not supported as an open conjecture by the supplied status evidence.