Groups whose proper subgroups of infinite rank have a permutability transitive relation

Let G be a group. A subgroup H of G is called permutable if HX = XH for all subgroups X of G. Permutability is not in general a transitive relation, and G is called a PT-group if, wheneverK is a permutable subgroup of G andH is a permutable subgroup of K, we always have that H is permutable in G. The property PT is not inherited by subgroups, and G is called a (PT) over bar -group if all its subgroups have the PT-property. We prove that if G is a soluble group of infinite rank whose proper subgroups of infinite rank have the (PT) over bar -property, then G is a PT-group.