Let $A$ be a supersingular abelian surface over a sufficiently large finite field $\mathbb{F}_q$ of characteristic $2$ and let $K_A = A/(-1)$ be the Kummer surface. Shioda ("Kummer surfaces in characteristic 2") and Katsura ("On Kummer surfaces in characteristic 2") proved that $K_A$ is rational.

There is the open conjecture about $\mathbb{F}_q$-unirationality of a rational surface with a smooth $\mathbb{F}_q$-point. See, for example, "Rational varieties: algebra, geometry and arithmetic" by Manin and Tsfasman.

Let $A$ be $\mathbb{F}_q$-simple, it est, $A$ isn't isogenous a direct product of 2 elliptic curves over $\mathbb{F}_q$. Is $K_A$ unirational over $\mathbb{F}_q$ in this case? In particular, what is $\mathbb{F}_q$-minimal smooth model for $K_A$?