Forbidden-minor conjecture for non-even regular oriented matroids

A regular oriented matroid is an oriented matroid whose underlying matroid is regular. For integers $m,n\geq 2$ with $m+n$ odd, let $\vec{K}_{m,n}$ denote the complete bipartite graph with its natural orientation, let $M^*(\vec{K}_{m,n})$ be its cographic oriented matroid, and let a $\mathsf{GB}$-minor be a minor obtained by the graphic-bond operations used in the paper. An oriented matroid is non-even when it is not even.

Forbidden-minor conjecture. A regular oriented matroid $M$ is non-even if and only if none of its $\mathsf{GB}$-minors is isomorphic to $M^*(\vec{K}_{m,n})$ for some $m,n\geq 2$ such that $m+n$ is odd.

The conjecture would identify the complete list of forbidden $\mathsf{GB}$-minors for non-even regular oriented matroids by combining the graphic and cographic cases. The paper reports computational evidence for the $10$-element regular matroid $R_{10}$; the general conjecture remains open, including the stated complexity problem for regular oriented matroids and the case of bond matroids.