The potential crossing pair characterization of graphs with crossing number at least 2

Let $G$ be a non-planar graph. A pair of edges $\{e,f\}$ of $G$ is a potential crossing pair if, for every Kuratowski subgraph $H$ of $G$, $e$ and $f$ form a crossing pair of $H$. The crossing number of $G$, denoted by $\operatorname{cr}(G)$, is the minimum number of crossings in a drawing of $G$ in the plane.

Potential crossing pair conjecture. The graph $G$ has crossing number at least $2$ if and only if it does not have a potential crossing pair.

This question arises from the characterization of crossing pairs in the preceding theorem: a potential crossing pair that is not separated by cycles is a crossing pair. Thus, any potential crossing pair in a graph with crossing number at least $2$ would necessarily be separated by cycles. The source does not provide a resolution of this characterization.