Connes' character rigidity conjecture for higher-rank lattices

Let $\Gamma$ be a higher-rank lattice and let a character mean the character object considered in the source. Connes' character rigidity conjecture. Every character of $\Gamma$ is either central or finite dimensional. The source attributes this conjecture to Connes, notes that it generalizes Margulis' Normal Subgroup Theorem, and records proofs in several cases; it gives no claim of a complete resolution.