Monotone e-embracing exchange sequences

Determine whether every pair of e-embracing bases in an oriented matroid admits a monotone e-embracing exchange sequence, in which the intersection with the target basis never decreases.

Background

The theorem for oriented graphic matroids establishes a stronger property than the basic distance bound: the constructed sequence is monotone, meaning that the size of the intersection with the target basis does not decrease at any step.

This motivates asking whether monotonicity holds for all oriented matroids. The question is posed as a strengthening of the main oriented-matroid conjecture and is not resolved by the paper, which only establishes the property for oriented graphic matroids.