Bohn, Faenza, Fiorini, Fisikopoulos, Macchia, and Pashkovich (2015) conjectured that 2-level polytopes cannot simultaneously have many vertices and many facets, namely, that the maximum of the product of the number of vertices and facets is attained on the cube and cross-polytope. This was proved in a recent work by Kupavskii and Weltge. In this paper, we resolve a strong version of the conjecture by Bohn et al., and find the maximum possible product of the number of vertices and the number of facets in a 2-level polytope that is not affinely isomorphic to the cube or the cross-polytope. To do this, we get a sharp stability result of Kupavskii and Weltge's upper bound on $\left|\mathcal A\right|\cdot\left|\mathcal B\right|$ for $\mathcal A,\mathcal B \subseteq \mathbb R^d$ with a property that $\forall a \in \mathcal A, b \in \mathcal B$ the scalar product $\langle a, b\rangle \in\{0,1\}$.

翻译：Bohn、Faenza、Fiorini、Fisikopoulos、Macchia和Pashkovich (2015)猜想，2-水平多面体不能同时具有大量顶点和大量面，即顶点数与面数乘积的最大值在立方体和交叉多面体上达到。这一猜想在Kupavskii和Weltge最近的一项工作中得到证明。在本文中，我们解决了Bohn等人提出的强版本猜想，并找到了非仿射同构于立方体或交叉多面体的2-水平多面体中顶点数与面数乘积的最大可能值。为此，我们得到了Kupavskii和Weltge关于$\left|\mathcal A\right|\cdot\left|\mathcal B\right|$上界的一个尖锐稳定性结果，其中$\mathcal A,\mathcal B \subseteq \mathbb R^d$满足性质：$\forall a \in \mathcal A, b \in \mathcal B$，标量积$\langle a, b\rangle \in\{0,1\}$。