Eells' conjecture on harmonic maps from the 3-sphere to the 2-sphere

Let $f:\mathbb{S}^3\to\mathbb{S}^2$ be a harmonic map, where both spheres have their round metrics. Let $\pi:\mathbb{S}^3\to\mathbb{S}^2$ be the Hopf fibration. Eells' conjecture. Every such map factors as

$$f=g\circ\pi,$$

where $g:\mathbb{S}^2\to\mathbb{S}^2$ is a conformal map. This remains open in general; the paper proves it under suitable conditions on the Hessian and singular values of $f$.