Existence of reflexive abelian groups of arbitrarily large cardinalities

A reflexive abelian group is an abelian group naturally isomorphic to its double dual. Existence conjecture. There are reflexive abelian groups of arbitrarily large cardinalities. This is the problem of whether reflexive abelian groups can have cardinality at least the first measurable cardinal; the preceding result shows that the standard direct-sum examples cease to be reflexive at and above that cardinal.