The coarse median structures conjecture for mapping class groups

Let $S$ be a non-sporadic finite-type surface, and let $\operatorname{MCG}(S)$ denote its mapping class group. A coarse median structure on a group is a coarse median operation compatible with a compatible word metric. Coarse median structures conjecture. The mapping class group $\operatorname{MCG}(S)$ has at least two coarse median structures. This would extend the paper's result for the mapping class group of the five-holed sphere and suggest that non-sporadic mapping class groups support genuinely distinct coarse geometric structures. The statement is presented as a future direction and is not resolved here.