Six-connectivity conjecture for global rigidity of 3-dimensional combinatorial zeolites

A 3-dimensional combinatorial zeolite is the line graph of a 4-regular graph. Let $G$ be such a graph, and suppose that $G$ is 6-connected, meaning that deletion of any set of at most five vertices leaves it connected.

Combinatorial-zeolite conjecture. Every 6-connected 3-dimensional combinatorial zeolite is globally rigid in $\mathbb{R}^3$.

The source motivates this by the analogous two-dimensional result, where 3-connected combinatorial zeolites are globally rigid in $\mathbb{R}^2$. It also notes that a stronger 4-connectivity conjecture was posed by others; the 6-connectivity statement remains open in the supplied text.