Symmetric tensegrity analogue of Connelly's theorem

Let $(\hat{G},p)$ be a fully symmetric tensegrity framework, with cables and struts whose self-stress coefficients are denoted by $\omega_{ij}$. A symmetry-preserving finite flex is a finite flex that preserves the framework's symmetry.

Symmetric tensegrity analogue of Connelly's theorem. If $(\hat{G},p)$ has a cable or strut and no symmetry-preserving finite flex, then it has a fully symmetric, non-zero, proper self-stress satisfying

$$\omega_{ij}>0$$

on cables and

$$\omega_{ij}<0$$

on struts.

This would extend Connelly's theorem to fully symmetric tensegrity frameworks. The source presents it as a conjectural symmetry adaptation of the basic proof; no resolution is given here.