Nowhere vanishing harmonic 1-forms on 3-manifolds

Consider $$(S^1 \times \Sigma^2, g)$$, where $$g$$ is any Riemannian metric on the compact and closed $$3$$-manifold $$S^1 \times \Sigma^2$$.

Question: Does there always exist a nowhere vanishing harmonic $$1$$-form on $$S^1 \times \Sigma^2$$? If the answer to this question is No, how about the generalisation to $$k$$-parameter families of metrics?

I am particularly interested in the case of $$\Sigma=S^1 \times S^1$$. This is a crosspost from https://math.stackexchange.com/questions/4262844/nowhere-vanishing-harmonic-1-forms-on-3-manifolds.