Let $X$ be a quasi-hyperelliptic surface in characteristic 3 where the canonical bundle $K_X$ is trivial.

**Question:** Is there a finite étale covering $Y \rightarrow X$ such that

- $Y$ is an abelian variety, or
- the étale fundamental group of $Y$ is trivial?

**Motivation:**

The Beauville-Bogomolov decomposition theorem for $\mathbb C$ dictates that a smooth projective variety $X/\mathbb C$ with $K_X$ trivial has a finite étale cover $Y \rightarrow X$ such that $Y$ is a product of abelian varieties, "strict" Calabi–Yau varieties and irreducible symplectic varieties. The two latter has their étale fundamental groups trivial.

Though there are no universally agreed definition of irreducible symplectic varieties in char. p, we could still "disprove" Beauville-Bogomolov decomposition in char. p if the covering spaces have undesirable properties.