Graphic–cographic conjecture on NBC bases

Let $M$ be a matroid on ground set $E(M)$. A standard ordering of $E(M)$ is an ordering satisfying the standard-ordering conditions defined for the matroid's broken-circuit complex. The complex $NBC(M)$ is the broken-circuit complex associated with $M$. Graphic–cographic NBC-basis conjecture. If $M$ is graphic or cographic, then there exists a standard ordering of $E(M)$ such that $NBC(M)$ has an NBC basis. This extends the conjecture attributed to Brown and Sagan from graphic matroids to graphic and cographic matroids.