Submodular-cut extension conjecture for oriented matroids

Let $\mathcal{O}$ be an oriented matroid with underlying matroid, and let $\mathcal{M}$ be a modular cut of that matroid such that there is an extension $\mathcal{O}\cup p$ corresponding to $\mathcal{M}$. Let $\mathcal{M}'\subseteq\mathcal{M}$ also be a modular cut.

Submodular-cut extension conjecture. There exists an extension $\mathcal{O}\cup p'$ corresponding to $\mathcal{M}'$.

The authors present this as a potentially useful conjecture for proving the preceding general-rank result and note that it is interesting in its own right. No resolution is supplied.