Finite cross-ratio conjecture for quaternionic unimodular matroids

Let $\mathbb{H}$ be the quaternions, with norm $|\cdot|$, and let $p,q,r\in\mathbb{H}$ satisfy

$$|i-j|=1$$

for all distinct $i,j\in\{0,1,p,q,r\}$. Let $M$ be a quaternionic unimodular matroid (QU matroid), and let $(\mathbb{H},\langle-1,p,q,r\rangle)$ be the skew partial field generated by these elements. Finite cross-ratio conjecture. The matroid $M$ is representable over the skew partial field $(\mathbb{H},\langle-1,p,q,r\rangle)$. This is presented as a concrete conjectural restriction on the cross ratios needed for representations of QU matroids.