Failure of the matroid matching property in odd-dimensional rigidity matroids

For integers $k\geq 2$ and sufficiently large $n$, consider the rigidity matroid ${\mathcal R}_{2k+1}(K_n)$ of the complete graph $K_n$ in dimension $2k+1$. The matroid matching conjecture. The matroid ${\mathcal R}_{2k+1}(K_n)$ does not have the matroid matching property for all $k\geq 2$ and sufficiently large $n$. The preceding proof establishes the corresponding even-dimensional result, while the odd-dimensional case is suspected to fail in this way for dimensions at least five and large complete graphs.