The coarse assembly map conjecture for warped cones

Let $G$ be a finitely generated group acting ergodically by probability-preserving Lipschitz homeomorphisms on a compact metric probability space $(M,\operatorname{dist},m)$, with $m$ upper uniform, and suppose the action has a spectral gap. Let $X=\mathcal{O}_G M$ be the associated warped cone, and let $C^*(X)$ be its Roe algebra. The coarse assembly map is the map

$$\lim_{d\to\infty}K_*(P_d(X))\longrightarrow K_*(C^*(X)).$$

The warped-cone assembly conjecture. The coarse assembly map is not surjective for the warped cone $X=\mathcal{O}_G M$. This would provide a geometric obstruction to the coarse Baum–Connes conjecture beyond the previously known expander examples.