Six-element-minor characterization of pure oriented matroids

Let ${\mathcal{M}}$ be an oriented matroid, and let ${\mathcal{M}}_1\leadsto {\mathcal{M}}_2$ denote a rank-preserving weak map. The graphical oriented matroids associated with the directed graphs $K_4$ and $K_{2,3}$ are denoted by ${\mathcal{M}}_{\vec K_4}$ and ${\mathcal{M}}_{\vec K_{2,3}}$.

Six-element-minor conjecture. The oriented matroid ${\mathcal{M}}$ is pure if and only if all of its six-element minors are pure. Equivalently, ${\mathcal{M}}$ is pure if and only if neither ${\mathcal{M}}_{\vec K_4}$ nor ${\mathcal{M}}_{\vec K_{2,3}}$ can be obtained from ${\mathcal{M}}$ by taking minors and rank-preserving weak maps. In particular, if ${\mathcal{M}}_1\leadsto {\mathcal{M}}_2$ is a rank-preserving weak map and ${\mathcal{M}}_1$ is pure, then ${\mathcal{M}}_2$ is pure as well.

This conjecture proposes a finite-obstruction characterization of purity and predicts that purity is preserved under rank-preserving weak maps. The source provides no resolution of the conjecture.