Uniqueness conjecture for the Clifford torus among CR torus minimizers

Let $S^3$ be the three-sphere, let $E_1$ be the energy functional considered in the source, and let $\Sigma_{\frac{\sqrt{2}}{2}}\subset S^3$ denote the Clifford torus. A surface of torus type has genus $1$. Two surfaces are regarded as equivalent when they are related by a $CR$ automorphism of $S^3$. Clifford-torus uniqueness conjecture. The Clifford torus $\Sigma_{\frac{\sqrt{2}}{2}}$ is a unique minimizer of $E_1$ among all surfaces of torus type, up to $CR$ automorphisms of $S^3$. The source verifies that the Clifford torus satisfies the Euler–Lagrange equation for $E_1$; the asserted global uniqueness and minimizing property remain conjectural.