Six-connectivity conjecture for rigidity of K4-covered graphs

Let $G$ be a graph that is 6-connected (it remains connected after deletion of any set of at most five vertices) and $K_4$-covered (every edge lies in a copy of $K_4$).

Six-connectivity conjecture. Every 6-connected $K_4$-covered graph is rigid in $\mathbb{R}^3$.

The paper proves rigidity under the stronger hypothesis of 5-connectivity together with $\mathcal{R}$-bridgelessness, and notes that the analogous cofactor statement is known. The conjecture would be false with 5-connectivity in place of 6-connectivity, so the stated bound is the proposed improvement.