The conjectural picture for supersingular symplectic varieties

Let $X$ be a symplectic variety defined over an algebraically closed field $k$, with vanishing odd Betti numbers. Call $X$ second-Artin supersingular if its second crystalline cohomology group is a supersingular $F$-crystal. The notions of fully Shioda supersingular, unirational, and rational Chow motive of Tate type are understood through the conclusions below.

Supersingularity conjecture. If $X$ is second-Artin supersingular, then:

1. $X$ is fully Shioda supersingular; equivalently, the

-adic and crystalline cycle class maps are surjective. 2. $X$ is unirational. 3. The rational Chow motive of $X$ is of Tate type. In particular, the

-adic and crystalline cycle class maps are injective.

This conjecture relates the cohomological, geometric, and motivic aspects of supersingular symplectic varieties, extending the expected relationships known for supersingular K3 surfaces. The supplied text does not state whether the conjecture has been resolved.