Shioda's unirationality conjecture for K3 surfaces

A K3 surface is a smooth projective surface with trivial canonical bundle and $h^1(S,\mathcal O_S)=0$. It is Shioda-supersingular when its Picard number equals its second Betti number:

$$\rho(S)=b_2(S).$$

Shioda's conjecture. A K3 surface is unirational if and only if it is Shioda-supersingular.

Unirational K3 surfaces and the known examples from Kummer and Fermat surfaces support this equivalence. The conjecture was known in characteristic $2$ through the classification of Shioda-supersingular K3 surfaces and the existence of quasi-elliptic fibrations, but the source does not state a complete resolution.