Complete-digraph edge-reversal conjecture for Jacobian matroids

Let $G_1$ and $G_2$ be complete digraphs on $p$ nodes, with all edge directions agreeing except for the edge between nodes $p-1$ and $p$. Complete-digraph edge-reversal conjecture. The two graphs have the same Jacobian matroid. The conjecture concerns a limitation of Jacobian-matroid methods: even complete graphs differing only by this edge orientation may define the same matroid. The paper reports computational evidence for related coincidences but gives no proof or resolution of this claim.