A volume inequality for convex polytopes with d+2 vertices on the sphere

Let $d\in\mathbb{N}$ and suppose $Q\subset\mathbb{S}^{d-1}$ is a convex polytope with $d+2$ vertices. Let $Q^o$ denote its polar polytope, let $\operatorname{vol}_k(Q)$ denote the vector of $k$-dimensional volumes associated with $Q$, and let $\varepsilon_d$ be a quadratic form satisfying

$$0\leq \varepsilon_d(Q)\leq C\operatorname{vol}_d(Q)^2.$$

The volume inequality. One has

$$\|\operatorname{vol}_d(Q) \|_{\ell_2}^2+\varepsilon_d(Q)\leq \frac{\left(\| \operatorname{vol}_1(Q)\|_{\ell_2}^2+\|\operatorname{vol}_d(Q^o) \|_{\ell_2}^{2}\right)^d}{ c_d\cdot d!^2\cdot d^d},$$

where $c_d=2(d+2)^{d-1}$.

The statement gives a quantitative geometric inequality for convex polytopes with $d+2$ vertices, with the error term controlled by the squared volume. Its status is unclear from the supplied context.