The quadratic Dirac-kernel estimate for two-spheres

Let ${\cal D}$ be a Dirac operator on a two-sphere, and let $U$ be its potential. Define

$$N=\dim_{\mathbb H}\operatorname{Ker}{\cal D}=\frac{1}{2}\dim_{\mathbb C}\operatorname{Ker}{\cal D}.$$

Quadratic Dirac-kernel estimate. For every Dirac operator on a two-sphere, the estimate

$$\int_M U^2\,dx\wedge dy\geq \pi N^2$$

should hold.

The source first proves this estimate for real-valued potentials depending on one variable and then states the all-operator version as a conjecture. It was later proved by Ferus, Leschke, Pedit, and Pinkall, so its database status is solved.