Caoduro–Khodamoradi–Paat–Shepherd's oriented-matroid embracing-distance conjecture

Let $M$ be an oriented matroid of rank $r$, let $e$ be the distinguished element defining the embracing condition, and let $A$ and $B$ be bases of $M$. Call them $e$-embracing when they satisfy the source's $e$-embracing basis condition, and let their $e$-embracing exchange distance be the minimum number of exchanges in a sequence of $e$-embracing bases joining them. Oriented-matroid embracing-distance conjecture. The $e$-embracing exchange distance of two $e$-embracing bases of a rank-$r$ oriented matroid is at most $r$. In the abstract oriented-matroid setting, it remains open even whether an exchange sequence satisfying the required embracing condition always exists, regardless of its length.