Finite asymptotic dimension conjecture for finite-rank coarse median spaces

A coarse median space is a metric space equipped with a ternary operation whose finite subsets can be approximated by finite median algebras with metric-controlled distortion. A space has finite asymptotic dimension if there is an integer $n$ such that, for every scale, it admits a uniformly bounded cover with multiplicity at most $n+1$. The paper considers geodesic coarse median spaces of finite rank and at most exponential volume growth, for which it has established subexponential asymptotic dimension growth.

Finite asymptotic dimension conjecture. Every geodesic coarse median space with finite rank has finite asymptotic dimension.

This would strengthen the stated result for finite-rank coarse median spaces with at most exponential volume growth. Coarse median spaces include hyperbolic spaces, mapping class groups, and CAT(0) cubical groups; the conjecture is presented as a general expected result, and its resolution is not specified in the source.