The real-rank conjecture for uniform Roe algebras

Let $X$ be a bounded geometry metric space, and let $C^*_u(X)$ denote its uniform Roe algebra. The real-rank conjecture. If $X$ has asymptotic dimension at least two, then $C^*_u(X)$ does not have real rank zero. The preceding discussion shows that this holds for word hyperbolic groups, while the conjecture is presented as natural in the broader class of bounded geometry metric spaces of asymptotic dimension at least two.