Connectivity characterization for Morse boundaries of Coxeter groups

A Coxeter group has a defining graph and a Morse boundary, the space of Morse geodesic directions; it is one-ended if it has one end and wide-avoidant if it satisfies the wide-avoidance condition for wide special subgroups.

Connectivity conjecture. A Coxeter group has connected, non-empty Morse boundary if and only if it is one-ended and wide-avoidant.

The paper proves that a Coxeter group that is not wide and not wide-avoidant has disconnected Morse boundary, together with a corresponding splitting result. The conjecture proposes the converse characterization of connected, non-empty Morse boundaries.