Existence of supersingular curves of genus five via unramified double covers

Let $p$ be an odd prime, and let $X$ be a smooth curve of genus $3$ over $\overline{\mathbf F}_p$. An unramified double cover of $X$ is a degree-two étale morphism $\pi:Y\to X$; by the Riemann–Hurwitz formula, $Y$ has genus $5$. A smooth curve is supersingular when its Jacobian is a supersingular principally polarized abelian variety. Supersingular genus-five curve conjecture. For every odd prime $p$, there exists a smooth curve $X$ of genus $3$ over $\overline{\mathbf F}_p$ with an unramified double cover

$$\pi:Y\to X$$

such that $Y$ is a supersingular curve of genus $5$. This would establish the existence of supersingular curves of genus $5$ in odd characteristic through the Prym construction described in the paper; the paper provides evidence for the conjecture, while its general validity remains open.