The anticanonical bigness conjecture for projective splinters

Let $X$ be a splinter, meaning that every finite surjective morphism onto $X$ admits a splitting of the corresponding map of structure sheaves. Assume that $X$ is a $bbQ$-Gorenstein projective scheme over a field of positive characteristic.

Anticanonical bigness conjecture. If $X$ is a splinter, then $-K_X$ is big.

This conjecture is motivated by the expectation that splinters in positive characteristic should be globally $F$-regular; global $F$-regularity imposes strong positivity constraints, including bigness of the anticanonical divisor in the $bbQ$-Gorenstein case. Its resolution is not specified in the source.