I am reading the book "Graph Classes: A Survey", and I know that Perfect Graph Conjecture (PGC) and Strong Perfect Graph Conjecture (SPGC) once were famous conjectures about perfect graphs.
PGC and SPGC have been proved in 1972 and 2006, respectively. I want to know after 2006, are there some famous conjectures still open about perfect graphs (or subclasses of perfect graphs)?