Dirac eigenvalue-sum conjecture for genus-one surfaces in the three-sphere

Let $M^{2}\subset\mathbb{S}^{3}(1)$ be an embedded closed surface of genus one, and suppose that the curvature endomorphism satisfies $\Re|_{\slashed{E}}=0$. Let $\bar{\Gamma}_{1}$ and $\bar{\Gamma}_{2}$ be the first two nonzero eigenvalues of $\slashed{D}^{2}$ on $C^{\infty}(\slashed{E})$. Dirac eigenvalue-sum conjecture.

$$\frac{1}{2}\sum_{i=1}^{2}\bar{\Gamma}_{i}\geq\frac{4\pi^{2}}{\operatorname{area}(M^{2})}.$$

This proposed lower bound is closely related to the Willmore conjecture through the geometry of embedded genus-one surfaces in $\mathbb{S}^{3}(1)$. The supplied source does not state whether it has been resolved.