Non-Las-Vergnas implies total non-Euclideaness

Determine whether every oriented matroid that is not Las Vergnas is totally non-Euclidean, meaning that it has no Euclidean oriented matroid programs.

Background

The paper distinguishes Las Vergnas oriented matroids, which have an adjacent mutation at every element, from totally non-Euclidean oriented matroids, which have no Euclidean programs. It provides examples in both categories but does not establish whether failure of the Las Vergnas property forces total non-Euclideaness.