Appears in collection : Algebraic and finite groups, geometry and representations. 50 years of Chevalley seminar / Groupes algébriques, groupes finis, géométries, représentations : 50 ans de séminaire Chevalley

I will present arguably the most basic one among the set of conjectures stated in 1998 by Broue, Malle and Rouquier (following early work by Broue and Malle) about the generalized Iwahori-Hecke algebras associated to complex reflection groups. By a combination of several kind of arguments and lots of hand-writen as well as computer-assisted calculations, it seems that a complete proof is now within reach. I will report on recent progress by my PhD student E. Chavli, as well as on a recent work by G. Pfeiffer and myself on this topic.