Filtration-equivalent ℵ1-separable abelian groups of cardinality ℵ1

We show that it is consistent with ordinary set theory ZFC and the generalized continuum hypothesis that there exist two ℵ1-separable abelian groups of cardinality ℵ1 which are filtration-equivalent and one is a Whitehead group but the other is not. This solves one of the open problems from Eklof and Mekler (2002) [2].