A poset is a containment of paths in a tree (CPT) if it admits a representation by containment where each element of the poset is represented by a path in a tree and two elements are comparable in the poset if and only if the corresponding paths are related by the inclusion relation. Recently Alcón, Gudiño and Gutierrez [1] introduced proper subclasses of CPT posets, namely dually-CPT, and strongly-CPT. A poset P is dually-CPT, if and only if P and its dual P d both admit a CPT representation. A poset P is strongly-CPT, if and only if P and all the posets that share the same underlying comparability graph admit a CPT representation. Where as the inclusion between Dually-CPT and CPT was known to be strict. It was raised as an open question by Alcón, Gudiño and Gutierrez [1] whether strongly-CPT was a strict subclass of dually-CPT. We provide a proof that both classes actually coincide.