Česnavičius's compactification conjecture for isotrivial reductive groups

Let $S$ be a Noetherian scheme and let $G$ be an isotrivial reductive group over $S$, meaning that $G$ becomes split after a finite étale cover of $S$. A $G$-equivariant $S$-fiberwise dense open immersion is an open immersion whose restriction to every fiber of $S$ is dense and which is compatible with the left action of $G$. Česnavičius's compactification conjecture. There are a projective, finitely presented $S$-scheme $\overline{G}$ equipped with a left $G$-action and a $G$-equivariant $S$-fiberwise dense open immersion

$$G \hookrightarrow \overline{G}.$$

This conjecture asks for an equivariant projective compactification of every isotrivial reductive group scheme. The paper's abstract states that the conjecture is verified by constructing a smooth projective compactification; the supplied status is unknown, so the database status is left open pending confirmation of the resolution.