The freeness ordering of generic rigidity matroids

Let $n\ge d$ and let $\mathcal{H}_d(n)$, $\mathcal{R}_d(n)$, and $\mathcal{C}_{d-1}^{d-2}(n)$ denote the generic hyperconnectivity, bar-and-joint, and cofactor rigidity matroids, respectively. For matroids on the same ground set, write $\mathcal{M}\le\mathcal{N}$ when every independent set of $\mathcal{M}$ is independent in $\mathcal{N}$.

Kalai–Whiteley conjecture. For every $n$ and $d$,

$$\mathcal{H}_d(n)\le\mathcal{R}_d(n)\le\mathcal{C}_{d-1}^{d-2}(n).$$

This compares the relative freeness of the three principal generic rigidity matroids. It holds for $d=1$ and $d=2$, while the comparison is open in higher dimensions in general; in particular, the left inequality is open even for $d=3$.