Global surface-area maximizer conjecture for seven-vertex inscribed polytopes

Let $\mathcal{I}_7$ denote the class of seven-vertex polytopes inscribed in the unit sphere $\mathbb{S}^2$, and let $S(P)$ be the surface area of $P$. Let $e_3$ be the third standard basis vector, and let $\operatorname{span}(e_3)^\perp$ be the equatorial plane. Global surface-area maximizer conjecture. For every $P\in\mathcal{I}_7$,

$$S(P) \leq \frac{5}{4}\sqrt{50-6\sqrt{5}}=7.560546\ldots$$

with equality if and only if $P$ is a pentagonal bipyramid with two vertices at the poles $\pm e_3$ and the other five forming an equilateral pentagon in the equator $\mathbb{S}^2\cap \operatorname{span}(e_3)^\perp$. Theorem 7 establishes this bound and equality characterization for $P\in\mathcal{M}_7$; the conjecture asserts that the same pentagonal bipyramid is the global maximizer in the larger class $\mathcal{I}_7$.