General Dirac-potential integral conjecture for spheres

Let $\mathcal E$ be the spinor bundle over a sphere, let $\mathcal D$ be a Dirac operator on $\mathcal E$, and write $U(z,\bar z)$ for its potential in a Weierstrass representation. Let $N=\dim_{\mathbf H}\operatorname{Ker}\mathcal D$, where the kernel is viewed as a quaternionic vector space. General Dirac-potential conjecture. The estimate

$$\int_{\Sigma} U^2(z,\bar z)\,dx\wedge dy\geq \pi N^2$$

holds for any Dirac operator on $\mathcal E$. The estimate is derived in the paper for Dirac operators with one-dimensional potentials and is conjectured to remain valid for general Dirac operators on the sphere.