Bounded embracing exchange distance in oriented matroids

Prove that the e-embracing exchange distance between any two e-embracing bases of a rank-r oriented matroid is at most r.

Background

An e-embracing basis is a basis whose anchored fundamental circuit for an element e has positive sign on every basis element. An e-embracing exchange sequence transforms one e-embracing basis into another through single basis exchanges while preserving this property at every intermediate basis.

The conjecture extends the geometric 0-embracing simplex problem to arbitrary oriented matroids. The paper proves the bound for oriented graphic matroids of directed graphs, but leaves the general oriented-matroid case unresolved; indeed, it observes that even the existence of any such exchange sequence is not known in general.