Let X be a Fano variety. Dubrovin’s conjecture predicts, among other things, an equivalence between the semi-simplicity of QH(X) the big quantum cohomology of X and the existence of a full exceptional collection in D(X) the bounded derived category of X. Recently, Kuznetsov and Smirnov proposed a refinement of this conjecture if qH(X) the small quantum cohomology is not semi-simple. I will present this conjecture, discuss it in the case of coadjoint varieties and make connections with roots systems and simple surface singularities. This is a joint work with Maxim Smirnov