The integral isomorphism conjecture for bounded cohomology and uniformly finite homology

Let $M$ be a complete Riemannian manifold of dimension $n$ with a properly discontinuous, cocompact, isometric action by a group $G$, and let

$$\widehat{H}^n(M)\cong\ell^\infty(G)_G$$

be the isomorphism obtained from the integral described earlier in the paper, where $\ell^\infty(G)_G$ denotes the coinvariants of bounded functions on $G$. The integral isomorphism conjecture. The integral is an isomorphism.

The conjecture proposes an explicit description of the isomorphism between the bounded cohomology group $\widehat{H}^n(M)$ and the degree-zero uniformly finite homology group, identified with $\ell^\infty(G)_G$. The supplied text gives no evidence that the conjecture has been resolved.