Artin–Shioda conjecture on unirationality of supersingular K3 surfaces

Let $k$ be an algebraically closed field, and let $X$ be a $K3$ surface over $k$. The surface $X$ is supersingular if its Picard number is $22$, equivalently its Picard number equals its second Betti number. A surface $S$ is unirational if there exists a dominant rational map

$$\mathbb{P}^2\dashrightarrow S.$$

Artin–Shioda conjecture. Every supersingular $K3$ surface is unirational.

Shioda proved that every smooth projective unirational surface is supersingular, and Artin and Shioda conjectured the converse for $K3$ surfaces. The supplied paper proves the conjecture for supersingular $K3$ surfaces in characteristic $5$ with Artin invariant at most $3$; the general case is not resolved in the supplied context.