The London–Paris conjecture for cocircuit paths on topes

Let $\operatorname{\mathcal{M}}$ be a uniform oriented matroid. Let $\operatorname{\mathcal{C}}^*(\operatorname{\mathcal{M}})$ denote its cocircuits, and let $X,Y\in\operatorname{\mathcal{C}}^*(\operatorname{\mathcal{M}})$ be cocircuits that are vertices of at least one tope of $\operatorname{\mathcal{M}}$. For a tope $\operatorname{\mathcal{T}}$, write $d_{\operatorname{\mathcal{T}}}(X,Y)$ for the shortest-path distance between $X$ and $Y$ in the subgraph induced by the vertices of $\operatorname{\mathcal{T}}$.

London–Paris conjecture. There exists a tope $\operatorname{\mathcal{T}}$ in $\operatorname{\mathcal{M}}$ such that $X,Y\in\operatorname{\mathcal{T}}$ and

$$d_{\operatorname{\mathcal{M}}}(X,Y)=d_{\operatorname{\mathcal{T}}}(X,Y).$$

If true, this would imply a quadratic upper bound on the diameter of polytopes via the improved quadratic bound for cocircuit graphs. The conjecture has been verified computationally for oriented matroids with up to nine elements, but remains open in general.