Montiel–Urbano conjecture for the Clifford torus in the complex projective plane

Let $\mathbb{C}P^2$ be the complex projective plane, let $\mathcal{W}$ be the Willmore functional, and let $\mathcal{W}^-$ denote its component arising from the twistor decomposition. The Clifford torus is

$$T=\Big\{[z_1,z_2,z_3]\in\mathbb{C}P^2\,\Big|\,|z_1|=|z_2|=|z_3|\Big\}.$$

Montiel–Urbano conjecture. The Clifford torus achieves the minimum of the functional $\mathcal{W}^-$, and hence of $\mathcal{W}$, either among all tori in $\mathbb{C}P^2$ or among all Lagrangian tori in $\mathbb{C}P^2$.

The conjecture concerns the global minimization of the conformally invariant Willmore functional in the complex projective plane. The paper establishes strict Willmore stability of the Clifford torus, providing evidence, but does not resolve the global minimization claim.