The symmetry characterization conjecture for stable solutions

Let $(M,A,f)$ be the model space, and call a solution $\psi$ stable when the first variation of its length is zero and the second variation is positive. A geometrical symmetry is a transformation preserving the relevant structure and solution.

Stable-symmetry characterization conjecture. Stable solutions are characterized by geometrical symmetries: there is a group of symmetries acting on the space that preserves the solution.

The claim proposes that stability should reflect geometric symmetry, but the source gives no definition of the symmetry group or proof of the characterization.