Kim–Kim's bounded fundamental class conjecture for pinched negatively curved manifolds

Let $M$ be a pinched negatively curved manifold of infinite volume, of dimension $n$, and let $[hat{\omega}]in H^{n}_b(M,\mathbb{R})$ denote its bounded fundamental class. Kim–Kim's conjecture. The bounded fundamental class $[hat{\omega}]in H^{n}_b(M,\mathbb{R})$ vanishes if and only if the Cheeger isoperimetric constant $h(M)>0$. This conjecture extends the known equivalence for rank-one locally symmetric spaces and for pinched negatively curved $3$-manifolds with bounded geometry; the general case remains open.