A minor characterization of signed graphs with signed Colin de Verdière parameter at most three

A signed graph is a pair $(G,\Sigma)$ consisting of a graph $G$ and a set $\Sigma$ of odd edges. Let $\nu(G,\Sigma)$ denote its signed Colin de Verdière parameter. Write $K_5^o$ for the signed complete graph on five vertices with the specified odd-edge pattern, and $K_4^=$ for the corresponding signed complete graph on four vertices. A $\Delta Y$-transformation replaces a triangle by a new degree-three vertex adjacent to its vertices, and a $Y\Delta$-transformation is the reverse operation.

Minor characterization conjecture. A signed graph $(G,\Sigma)$ has $\nu(G,\Sigma)\leq 3$ if and only if $(G,\Sigma)$ has no minor isomorphic to $K_5^o$, $K_4^=$, or any signed graph $(H,\Omega)$ that can be obtained from $K_4^=$ by a sequence of $\Delta Y$- and $Y\Delta$-transformations.

This conjecture seeks the next excluded-minor characterization for the signed Colin de Verdière parameter after the established characterization for parameter at most two. The paper notes that $\nu(K_5^o)=4$ and $\nu(K_4^=)=4$, and that the parameter remains four under the indicated transformations; the proposed characterization is otherwise unresolved in the supplied text.