Whiteley's conjecture equating the 3-dimensional rigidity and cofactor matroids

For each integer $n\geq 2$, let $\mathcal{R}_3(K_n)$ denote the 3-dimensional rigidity matroid of the complete graph and let $\mathcal{C}_2^1(K_n)$ denote the corresponding cofactor matroid.

Whiteley's conjecture. These matroids are equal for every $n\geq 2$:

$$\mathcal{R}_3(K_n)=\mathcal{C}_2^1(K_n).$$

If true, this would transfer the known rank formula for the cofactor matroid to the 3-dimensional rigidity matroid and would imply Dress's conjecture. The source describes it as a long-standing conjecture and gives no resolution.