The hyperbolic asymptotic Cantor manifold conjecture

An $n$-asymptotic Cantor manifold is a metric space of asymptotic dimension $n$ that cannot be coarsely separated by a subset of asymptotic dimension at most $n-2$. For each integer $n\geq 2$, let $\mathbb{H}^n$ denote $n$-dimensional hyperbolic space. Hyperbolic asymptotic Cantor manifold conjecture. Every $n$-dimensional hyperbolic space, $\mathbb{H}^n$, is an $n$-asymptotic Cantor manifold, where $n\geq 2$. This asserts that hyperbolic spaces have the large-scale separation properties expected of asymptotic analogues of Cantor manifolds; the supplied text does not indicate whether the conjecture has been resolved.