Tran's conjecture on totally disconnected Morse boundaries of right-angled Coxeter groups

Let $\Lambda$ be a graph, and let

$$W_{\Lambda}=\left\langle V(\Lambda)\mid \{v^2\}_{v\in V(\Lambda)}\cup\{[v,w]\}_{\{v,w\}\in E(\Lambda)}\right\rangle$$

be its associated right-angled Coxeter group. A finitely generated group has a quasi-isometry invariant Morse boundary, denoted $\partial_M\Gamma$. An induced cycle in $\Lambda$ is burst if it contains a pair of non-adjacent vertices that are contained in an induced $4$-cycle. Tran's conjecture. The Morse boundary $\partial_M W_{\Lambda}$ is totally disconnected if and only if every induced cycle of length at least four in $\Lambda$ is burst. The classification of right-angled Coxeter groups with totally disconnected Morse boundary is open; this conjecture proposes a graph-theoretic characterization of precisely those groups.