Matroid matching property of the three-dimensional rigidity matroid

Determine whether the generic three-dimensional rigidity matroid \(\mathcal{R}_3\) has the matroid matching property.

Background

The paper establishes failure of the matroid matching property in even dimensions at least four, while the conjectured odd-dimensional result concerns dimensions at least five. Dimension three therefore remains unresolved.

The double circuit property and matroid matching property are related by a min-max theorem, but the paper does not determine either property for R3\mathcal{R}_3.

References

It remains unclear whether $_3$ has the double circuit property or the matroid matching property.

— $k$-fold circuits and coning in rigidity matroids  (2508.18838 - Hewetson et al., 26 Aug 2025) in Immediately after Conjecture 4.2 in Section 4