Bérczi–Nádor conjecture on e-embracing exchange distance

Let $\mathcal{M}=(E,\mathcal{C})$ be an oriented matroid of rank $r$, let $e\in E$, and let $\mathcal{C}^+$ denote its positive circuits. An $e$-positive circuit is a positive circuit $X\in\mathcal{C}^+$ with $X(e)=+$. For a basis $B$ with $e\notin B$, let $C(B,e)$ be the fundamental circuit whose support is contained in $B\cup\{e\}$ and whose $e$-coordinate is positive. The basis $B$ is $e$-embracing when $C(B,e)$ is $e$-positive. The $e$-embracing exchange distance of two $e$-embracing bases is the minimum length of a sequence of $e$-embracing bases in which consecutive bases differ by one exchange. Bérczi–Nádor's conjecture. The $e$-embracing exchange distance of any two $e$-embracing bases of a rank-$r$ oriented matroid is at most $r$. The paper's abstract states that this conjecture is confirmed for Lawrence oriented matroids, while the general conjecture is disproved by the results of the paper.