Isomorphism between the three-dimensional rigidity and cofactor matroids

Determine whether the generic three-dimensional rigidity matroid \(\mathcal{R}_3\) is isomorphic to the \(C_2^1\)-cofactor matroid.

Background

The unresolved double circuit property for $\mathcal{R}_3$ may be approachable through the closely related $C_2^1$-cofactor matroid. The paper notes that the rank function of the cofactor matroid has recently been characterized.

An isomorphism between the two matroids would transfer structural information between the cofactor and rigidity settings and could provide a route toward resolving the three-dimensional double circuit problem.