Artin–Rudakov–Shafarevich–Shioda conjecture on supersingular K3 surfaces

Let $k$ be an algebraically closed field of characteristic $p>0$. A K3 surface over $k$ is supersingular if the slope $[0,1)$ part of its rigid cohomology vanishes. It is unirational if there is a dominant rational map from some projective space to it.

Artin–Rudakov–Shafarevich–Shioda conjecture. A K3 surface over $k$ is supersingular if and only if it is unirational.

This conjecture concerns the relationship between the crystalline or rigid-cohomological notion of supersingularity and unirationality in positive characteristic. It is presented as a motivating conjecture for the paper's study of supersingular and rationally chain connected threefolds.