The symmetry-group correspondence conjecture for stable solutions

Let $(M,A,f)$ be a model space with stable solutions $\psi$, where stability means that the first variation of length is zero and the second variation is positive. Let symmetry groups act on $(M,A,f)$ and preserve the relevant structure.

Symmetry-group correspondence conjecture. There is a one-to-one correspondence between symmetry groups acting on $(M,A,f)$ and stable solutions for $\psi$ in $(M,A,f)$.

This strengthens the preceding qualitative symmetry characterization to a bijective correspondence. The source provides no construction of the correspondence or evidence that it is valid.