Characterization of separating cuts preserving the pfaffian number

Let a separating cut be a cut in a matchable graph whose contraction produces a graph, and let $\operatorname{pf}(G)$ denote the pfaffian number of a graph $G$. Separating-cut preservation problem. Characterize the separating cuts whose contractions preserve the pfaffian number.

The supplied text describes this as an interesting open problem. Such a characterization would identify precisely when contracting a separating cut leaves the pfaffian number unchanged.