Étale Reconstruction for ${\mathbb{F}}_p(t)$-Schemes

Voevodsky proved that normal schemes of finite type over finitely generated fields of characteristic $0$ can be reconstructed from their étale sites. Let $K$ be a field that is finitely generated over ${\mathbb{F}}_p(t)$. Grothendieck conjectured that perfections of finite type $K$-schemes can be reconstructed from their étale sites. Adapting Voevodsky’s methods, we prove this.