The weak stability conjecture for Willmore tori

A Willmore torus is stable when its Willmore second variation is nonnegative, and it is isothermic when it admits the corresponding isothermic structure in conformal surface geometry. Weak stability conjecture. Every stable Willmore torus is isothermic. The source presents this as a weaker version of the stability conjecture and notes that an affirmative answer would provide an alternative proof of the Willmore conjecture; it remains open in the supplied text.