  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
Alc\'on, Gudi\~{n}o and Gutierrez introduced proper subclasses of CPT posets,
namely dually-CPT, and strongly-CPT. A poset $\mathbf{P}$ is dually-CPT, if and
only if $\mathbf{P}$ and its dual $\mathbf{P}^{d}$ both admit a CPT
representation. A poset $\mathbf{P}$ is strongly-CPT, if and only if
$\mathbf{P}$ 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 Alc\'on,
Gudi\~{n}o and Gutierrez whether strongly-CPT was a strict subclass of
dually-CPT. We provide a proof that both classes actually coincide.
