The higher-degree comparison-map conjecture for semisimple Lie groups

Let $G$ be a connected semisimple Lie group without compact factors and with finite center. The comparison map

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

is the map from continuous bounded cohomology to continuous cohomology.

Higher-degree comparison-map conjecture. The comparison map

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

is an isomorphism.

This conjecture seeks a higher-degree analogue of the degree-two result of Burger and Monod. The degree-two comparison map is known to be an isomorphism under these hypotheses, while the corresponding assertion in all degrees is not established by the methods discussed here.