Higher-rank non-equivalence of intersection properties

Determine whether the three intersection properties for oriented matroids fail to be equivalent in every rank at least five.

Background

For matroids, non-equivalence is known in the higher-rank setting. The paper does not know whether the relevant counterexamples can be oriented, so it leaves open the oriented-matroid version.

References

Are the three intersection properties not equivalent for oriented matroids in rank $\ge 5$? Again this is shown for the matroid case. We do not know if the counterexamples given there are orientable.

— Mutations and (Non-)Euclideaness in oriented matroids  (2501.12951 - Wilhelmi, 22 Jan 2025) in Concluding remarks, Section concluding remarks