The crabbed-path strengthening for cocircuit distances

Let $\operatorname{\mathcal{M}}$ be a uniform oriented matroid, and let $X,Y\in\operatorname{\mathcal{C}}^*(\operatorname{\mathcal{M}})$ be cocircuits. Let $S(X,Y)$ denote the set used to define crabbed paths, and suppose $S(X,Y)=\emptyset$. A path is crabbed from $X$ to $Y$ when it satisfies the corresponding sign-coordinate restriction throughout.

Crabbed-path conjecture. There exists a crabbed path from $X$ to $Y$ whose length is no bigger than the length of any path from $X$ to $Y$ in $\operatorname{\mathcal{M}}$.

This is presented as a strengthening of the London–Paris conjecture. The surrounding discussion explains that the related tope-path conjecture would imply a quadratic bound on polytope diameters, while the linear diameter conjecture is already refuted; the crabbed-path strengthening itself is not resolved in the paper.