The intertwining conjecture for pivot-minors

A pivot-minor $H$ of a graph $G$ is proper if $|V(H)|<|V(G)|$. A graph $G$ is an intertwine of graphs $H_{1}$ and $H_{2}$ for pivot-minors if it contains both $H_{1}$ and $H_{2}$ as pivot-minors and no proper pivot-minor of $G$ contains both $H_{1}$ and $H_{2}$ as pivot-minors. Intertwining conjecture for pivot-minors. For graphs $G_{1}$ and $G_{2}$, there are only finitely many intertwines of $G_{1}$ and $G_{2}$ for pivot-minors. Together with the linking theorem, this conjecture would imply the stated connectivity corollary without an explicit function; it is presented as a motivation for the authors' theorem and remains unresolved in the supplied text.