Producing supersingular curves of genus five

For a prime $p$ congruent to three modulo four, we prove that there exists a smooth curve of genus five in characteristic $p$ that is supersingular. We produce this curve as an unramified double cover of a curve of genus three. We conjecture that the setting of unramified double covers of curves of genus three also produces supersingular curves of genus five when $p$ is congruent to one modulo four, and we computationally verify this conjecture for primes less than $100$. These results can be viewed as a generalization of work of Ekedahl and of Harashita, Kudo, and Senda.