Obstructions to unirationality for product-quotient surfaces over ${\bar{\mathbb{F}}}_p$

We construct a surface over ${\bar{\mathbb{F}}}_p$ with ${\pi }_1^{\acute{e}t}(X)=1$ that is supersingular – in the sense that $H_{\acute{e}t}^2(X,{\mathbb{Q}}_{\mathrm{\ell }}(1))$ is spanned by algebraic cycles – but is not unirational. This provides a counterexample to a 1977 conjecture of Shioda. To achieve this, we produce new obstructions to unirationality for product-quotient surfaces.