Infinite-dimensional bounded cohomology conjecture for mapping class groups

Let ${\cal M}_g$ be the mapping class group and let $H_b^2({\cal M}_g)$ denote its second bounded cohomology. Bounded-cohomology conjecture. For every genus $g$, the comparison map

$$H_b^2({\cal M}_g)\longrightarrow H^2({\cal M}_g;{\mathbb R})$$

is not injective; more strongly, $H_b^2({\cal M}_g)$ is infinite dimensional. The source motivates this by the known non-trivial bounded class in genus $2$ and results for hyperbolic groups, but gives no resolution of the assertion for all genera.