The theta-class strengthening of Murty's conjecture for Mandel oriented matroids

Let $G$ be a Mandel oriented matroid of rank $r$, and let $E$ be any $\Theta$-class of its tope graph. Theta-class strengthening of Murty's conjecture. The $\Theta$-class $E$ is incident to a vertex of an antipodal graph that is the Cartesian product of an edge and an antipodal graph of rank $r-2$. The source presents this as a suspected strengthening and does not report a resolution.