Ingleton's obstruction excluded-minor conjecture for base-orderability

Let $\Delta$ be a critical graph, let $P$ and $Q$ be the sets used to define the cyclic-flat lattice $\mathcal{Z}_\Delta^{P,Q}$, and let $M(\Delta)$ be the matroid with lattice of cyclic flats equal to $\mathcal{Z}_\Delta^{P,Q}$. An obstruction is the configuration defined in the source's preceding construction.

Obstruction excluded-minor conjecture. If $\Delta$ has an obstruction, then $M(\Delta)$ is an excluded minor for $\mathcal{BO}$.

The authors state this conjecture cautiously because Ingleton did not specify the construction of $M(\Delta)$ when $\Delta$ has an obstruction, and they cannot be certain that this is the matroid he intended. The assertion remains open.