\'Etale 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 \'etale 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 \'etale sites. Adapting Voevodsky's methods, we prove this.