Strong hull property for Cayley graphs of Coxeter groups

Let $W$ be any Coxeter group, and let $\operatorname{Cay}(W)$ be its undirected right Cayley graph with respect to the simple reflections. For vertices $u,v,w$, let $\operatorname{Conv}(u,v,w)$ denote their convex hull in this graph. The strong hull property is the inequality

$$\left|\operatorname{Conv}(u,v)\right|\cdot\left|\operatorname{Conv}(v,w)\right|\geq\left|\operatorname{Conv}(u,v,w)\right|.$$

Strong hull property theorem. The graph $\operatorname{Cay}(W)$ has the strong hull property.

This is stated as a main result and establishes the stronger inequality for every Coxeter group, including infinite ones. Consequently, the associated hull metric is available whenever the relevant hull inequalities define the metric.