Carvalho–Lucchesi–Murty laminar ELP cut conjecture

Let $G$ be a matching covered graph, and let $C$ be a nontrivial tight cut of $G$. An ELP cut is a barrier-cut or a 2-separation cut.

Carvalho–Lucchesi–Murty conjecture. The graph $G$ has a nontrivial ELP cut that is laminar with $C$.

If true, this would imply that every nontrivial tight cut of $G$ is ultimately an ELP cut in an ELP cut decomposition. The statement is presented as a conjecture in the source; its resolution is not established by the supplied text.