The stability conjecture for Willmore tori

A Willmore torus is called stable when the second variation of the Willmore functional is nonnegative, equivalently when the Jacobi operator satisfies

$$\delta^2\mathcal W\geq 0.$$

The Clifford torus is the standard homogeneous Willmore torus. Stability conjecture. The only stable Willmore torus is the Clifford torus. The source cites evidence from spectral-genus classifications and establishes the claim for Hopf Willmore tori, but does not resolve it for all Willmore tori.