Triviality conjecture for the bounded cohomology of the stable mapping class group

Let $\Gamma_{\infty}$ be the stable mapping class group, obtained as the direct limit of mapping class groups under the standard stabilization maps. Its bounded cohomology $H_b^*(\Gamma_{\infty})$ is the bounded cohomology of this group.

Stable bounded-cohomology conjecture. The bounded cohomology of $\Gamma_{\infty}$ is trivial:

$$H_b^*(\Gamma_{\infty})=0.$$

The authors note that the image of the comparison map $H_b^*(\Gamma_{\infty})\to H^*(\Gamma_{\infty})$ is trivial and, in particular, that $H_b^2(\Gamma_{\infty})=0$. They present the full triviality claim as a conjecture, while acknowledging that there is no more compelling evidence for it.