The solid Pfaffian graph matching-colouring conjecture

A matching covered graph is a graph in which every edge belongs to a perfect matching. A matching covered graph is solid if every non-trivial separating cut is tight, and Pfaffian if it has the Pfaffian property used in the paper. Let $G$ be a solid Pfaffian graph and let $M$ be a perfect matching of $G$; write $f0aMchromatic{G}{M}$ for the corresponding $M$-chromatic number.

Solid Pfaffian conjecture.

$$\Mchromatic{G}{M}\leq 2.$$

This is proposed as an extension of the bipartite Pfaffian result to non-bipartite matching covered graphs. The relevant structural questions for solid graphs remain widely open.