Supersingular genus-six curves from double covers branched at two points

Let $p$ be an odd prime. Let $X'$ be a smooth curve of genus $3$ over $\overline{\mathbf F}_p$, and let $\pi':Y'\to X'$ be a double cover branched at two points. By the Riemann–Hurwitz formula, $Y'$ has genus $6$. Branched-cover supersingularity conjecture. For every odd prime $p$, there exist $X'$ and $\pi'$ as above such that $Y'$ is a supersingular curve of genus $6$. The conjecture is motivated by the dimension and codimension calculation for genus-three curves and by the analogous genus-four construction, but remains open in general.