Complex-holomorphic-data conjecture for Willmore surfaces

Let $f:M\to S^4$ be a Willmore surface, where $M$ is a Riemann surface of genus $g$, and let $\perp_f$ denote its normal bundle. Assume

$$|\operatorname{deg}\perp_f|>4(g-1).$$

Complex-holomorphic-data conjecture. Then $f$ is given by complex holomorphic data.

The preceding lemma proves this when the Willmore surface admits a dual Willmore surface, and the theorem establishes related cases for Willmore spheres and Willmore tori with non-trivial normal bundle. The conjecture asks for the same conclusion under the stated normal-bundle degree condition in general.