One-component conjecture for minimal non-Euclidean matroids

Prove or disprove that every minimal non-Euclidean oriented matroid has only one very strongly connected component in its cocircuit graph.

Background

For uniform minimal non-Euclidean oriented matroid programs, the paper characterizes properties shared by every very strongly connected component, including the involvement of every ground-set element. It then proposes the stronger unresolved claim that there is only one such component.