The graphicness conjecture for 3-connected orderable binary matroids

A matroid $M$ is orderable if it admits an ordering of its circuits with the relevant ordering property, and it is 3-connected and binary in the usual matroid-theoretic senses. Graphicness conjecture. A $3$-connected orderable binary matroid is graphic. Although there are orderable binary matroids that are not graphic, no counterexample is known; the conjecture would distinguish graphic matroids within the class of binary matroids.