A proof of Generalized Connected Wedge Theorem

In the context of asymptotic $2$-to-$2$ scattering process in AdS/CFT, the Connected Wedge Theorem identifies the existence of $O(1\mathrm{/}{G}_N)$ mutual information between suitable boundary subregions, referred to as decision regions, as a necessary but not sufficient condition for bulk-only scattering processes, i.e., nonempty bulk scattering region $S_0$. Recently, Liu and Leutheusser proposed an enlarged bulk scattering region $S_E$ and conjectured that the non-emptiness of $S_E$ fully characterizes the existence of $O(1\mathrm{/}{G}_N)$ mutual information between decision regions. Here, we provide a geometrical or general relativity proof for a slightly modified version of their conjecture.