Artin--Shioda--Rudakov--Safarevic conjecture for K3 surfaces

Let $S$ be a K3 surface over an algebraically closed field of characteristic $p>0$. Supersingularity may be understood in the Artin or Shioda sense; these notions coincide for K3 surfaces.

Artin--Shioda--Rudakov--Safarevic conjecture. A K3 surface is supersingular if and only if it is unirational.

Unirational K3 surfaces are known to be supersingular, and the converse has been confirmed over fields of characteristic $2$ by Rudakov--Shafarevich via quasi-elliptic fibrations. The general assertion is presented here as resolved by the supplied status evidence, although the evidence sentence specifically records the characteristic-$2$ case.