Nonexistence of singular -harmonic 1-forms when the representation variety is zero-dimensional

Let $Y$ be a closed oriented $3$-manifold, and let $\mathcal R(Y)$ denote its relevant representation variety. A $\mathbb Z_2$-harmonic $1$-form on $Y$ is a triple $(\mathcal Z,\ell,\nu)$, where $\mathcal Z$ is its singular zero locus; the case $\mathcal Z=\emptyset$ consists of classical harmonic forms. Suppose that $\mathcal R(Y)$ is zero-dimensional.

Nonexistence conjecture. There exist no $\mathbb Z_2$-harmonic $1$-forms on $Y$ with $\mathcal Z\neq\emptyset$ with respect to any metric. In particular, there exist no $\mathbb Z_2$-harmonic $1$-forms on $S^3$, and no $\mathbb Z_2$-harmonic $1$-forms on $S^1\times S^2$ or $T^3$ except for the classical harmonic forms with $\mathcal Z=\emptyset$.

This conjecture distinguishes the behavior of $\mathbb Z_2$-harmonic $1$-forms from the spinor case. The supplied text gives no resolution, so the conjecture remains open.