Nonexistence of \(\mathbb Z/2\)-harmonic one-forms on every metric on \(S^3\)

Prove that for every Riemannian metric on the three-sphere \(S^3\), there is no nonzero \(\mathbb Z/2\)-harmonic one-form.

Background

The paper separately records a conjecture attributed to He–Wentworth–Zhang concerning the three-sphere. Unlike the paper’s main Seifert dichotomy, this statement concerns every Riemannian metric on $S^3$, rather than the existence of at least one metric without a nonzero $\mathbb Z/2$-harmonic one-form.

The conjecture is presented as an external unresolved claim and is distinct from the author’s conjecture for non-Haken hyperbolic rational homology three-spheres.