Bowden's bounded acyclicity conjecture for the stable mapping class group

Let $\Gamma_\infty$ denote the stable mapping class group, formed as the direct union of the groups of isotopy classes of diffeomorphisms of $\Sigma_{g,1}$ with compact support in the interior under stabilization by attaching a two-holed torus. A group is boundedly acyclic if its bounded cohomology with real coefficients vanishes in every positive degree:

$$H_b^n(\Gamma_\infty;\mathbb{R})=0\qquad\text{for every }n>0.$$

Bowden's conjecture. The stable mapping class group $\Gamma_\infty$ is boundedly acyclic.

Bowden's conjecture extends the known vanishing of stable commutator length and of $H_b^2(\Gamma_\infty;\mathbb{R})$ to all positive degrees. Its status is not resolved in the supplied source.