Restriction of maximal separated collections to mutation-closed domains

Let ${\mathcal{M}}$ be an oriented matroid on ground set $E$. A collection $\mathcal{S}\subseteq 2^E$ is ${\mathcal{M}}$-separated when it is separated according to the oriented-matroid separation relation. A mutation-closed domain ${{\mathcal{D}}}\subset 2^E$ is a union of connected components of the mutation graph of ${\mathcal{M}}$.

Restriction conjecture. If $\mathcal{S}$ is a maximal-by-size ${\mathcal{M}}$-separated collection inside $2^E$ and ${{\mathcal{D}}}$ is a mutation-closed domain, then $\mathcal{S}\cap {{\mathcal{D}}}$ is a maximal-by-size ${\mathcal{M}}$-separated collection inside ${{\mathcal{D}}}$.

This predicts that global maximal separated collections restrict to maximum separated collections on every mutation-closed domain. The source provides no resolution of the conjecture.