Inscribed 4-polytopes of maximal volume with seven and eight vertices

We prove, with sum-of-squares certificates checked in exact arithmetic, that every cyclic $4$-polytope with seven vertices inscribed in $\mathbb{S}^{3}$ has volume at most $2/5$. Hence no $4$-polytope of maximal volume with seven vertices inscribed in $\mathbb{S}^{3}$ is cyclic, answering a question