The non-crossing ELP-cut conjecture for matching-covered graphs

Let $G$ be a matching-covered graph. A tight cut is a cut met by exactly one edge of every perfect matching, and an ELP-cut is a barrier cut or a $2$-separation cut. A cut is nontrivial when neither shore is a singleton. Two cuts cross when each of the four intersections of their shores is nonempty.

Non-crossing ELP-cut conjecture. If $C$ is a nontrivial tight cut of $G$, then $G$ has an ELP-cut that does not cross $C$.

This conjecture was given as an alternative formulation related to the ELP Theorem, which asserts that every matching-covered graph with a nontrivial tight cut also has a nontrivial ELP-cut. The supplied text does not indicate whether the conjecture has been resolved.