Dupont's conjecture on the comparison map for continuous bounded cohomology

Let $G$) be a connected, non-compact, semi-simple Lie group with finite center. Write

$$c_G^*:H_{b,c}^*(G,\mathbb R)\rightarrow H_c^*(G,\mathbb R)$$

for the comparison map from continuous bounded cohomology to continuous cohomology. Dupont's conjecture. The map $c_G^*$ is always surjective. This conjecture is equivalent to the assertion that the relevant $G$-invariant forms on the associated symmetric space are bounded; its general validity is not established.