Monod's isomorphism conjecture for bounded cohomology of semisimple Lie groups

Let $G$ be a connected semi-simple Lie group with finite center. Let $H^*_{\rm cb}(G;\mathbb{R})$ denote its continuous bounded cohomology with real coefficients, and let $H^*_{\rm c}(G;\mathbb{R})$ denote its ordinary continuous cohomology.

Monod's isomorphism conjecture. The continuous bounded cohomology of $G$ with real coefficients is naturally isomorphic to its ordinary continuous cohomology:

$$H^*_{\rm cb}(G;\mathbb{R})\cong H^*_{\rm c}(G;\mathbb{R}).$$

This is presented as a stronger version of Dupont's conjecture and is open for all semisimple non-compact groups. The source attributes the problem to Monod.