Mandel totally non-Euclidean oriented matroids

Construct or determine whether there exist Mandel oriented matroids that are totally non-Euclidean.

Background

Mandelness requires a suitable Euclidean extension, whereas total non-Euclideaness prohibits Euclidean oriented matroid programs in the original oriented matroid. The paper asks whether these properties can coexist and does not provide an example.

References

On the other hand are there Mandel totally Non-Euclidean oriented matroids?

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