Gaetz–Gao's strong hull conjecture for Coxeter groups

Let $W$ be a Coxeter group, and let $\mathrm{Cay}(W)$ denote its undirected right Cayley graph associated with its generating set. For vertices $u,v,w\in V(\mathrm{Cay}(W))$, write $\mathrm{Conv}(u,v,w)$ for their convex hull, where convex hulls are taken with respect to graph distance. A graph has the strong hull property when

$$|\mathrm{Conv}(u,v)|\cdot|\mathrm{Conv}(v,w)|\geq|\mathrm{Conv}(u,v,w)|$$

for every three vertices $u,v,w$.

Gaetz–Gao's strong hull conjecture. Every Coxeter group $W$ has the property that its Cayley graph $\mathrm{Cay}(W)$ satisfies the strong hull property.

Gaetz and Gao proposed this as a conjecture about convexity in Cayley graphs of Coxeter groups. The paper containing the statement proves it for all affine irreducible Coxeter groups of rank $3$; the general case remains open based on the supplied source.