Monod's comparison-map conjecture for semisimple Lie groups

Let $G$ be a connected semisimple Lie group with finite center. Write $H^\bullet_{\mathrm{cb}}(G;\mathbb{R})$ for continuous bounded cohomology and $H^\bullet_{\mathrm{c}}(G;\mathbb{R})$ for continuous cohomology. Monod's conjecture. The natural comparison map

$$H^\bullet_{\mathrm{cb}}(G;\mathbb{R}) \to H^\bullet_{\mathrm{c}}(G;\mathbb{R})$$

is an isomorphism in all degrees. Surjectivity is known in many cases, while injectivity has only been established in the degrees and cases described in the source; the conjecture remains open in general. It predicts that the bounded cohomological dimension equals the dimension of the symmetric space associated to $G$, and is therefore positive and finite.