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 (i.e. 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.