Generic Newton points and the Newton poset in Iwahori-double cosets

We consider the Newton stratification on Iwahori-double cosets in the loop group of a reductive group. We describe a group-Theoretic condition on the generic Newton point, called cordiality, under which the Newton poset (that is, the index set for non-empty Newton strata) is saturated and Grothendieck's conjecture on closures of the Newton strata holds. Finally, we give several large classes of Iwahori-double cosets for which this condition is satisfied by studying certain paths in the associated quantum Bruhat graph.