Dress's rank formula conjecture for 3-dimensional rigidity matroids

Let $G$ be an $\mathcal{R}_3$-closed graph, let $\hat{\mathcal{X}}$ be the set of maximal cliques in $G$ of size at least five, and let $F$ be the set of edges of $G$ not covered by $\hat{\mathcal{X}}$. For a collection $\mathcal{X}$ of vertex subsets, write $\deg_{\mathcal{X}}(h)$ for the number of sets containing the hinge $h$, and let $\mathcal{H}(\mathcal{X})$ be the set of hinges, namely pairs with degree at least two.

The Dress Conjecture. The rank of $G$ satisfies

$$r_3(G)=|F|+\sum_{X\in \hat{\mathcal{X}}}(3|X|-6)-\sum_{h\in \mathcal{H}(\hat{\mathcal{X}})}(\deg_{\hat{\mathcal{X}}}(h)-1).$$

The conjecture proposes a rank formula for $\mathcal{R}_3$-closed graphs. The analogous formula is known for the cofactor matroid $\mathcal{C}_2^1$, providing evidence, while the conjecture remains open for the 3-dimensional rigidity matroid.