Existence of metrics with harmonic spinors on closed spin manifolds

Let $(M,s)$ be a closed spin manifold of dimension $m\ge 3$ with fixed spin structure $s$.

Harmonic-spinor existence conjecture. There exists a Riemannian metric $g$ on $M$ such that

$$h(M,g,s)>0.$$

The conjecture asks whether every closed spin manifold of dimension at least three admits a metric with a nonzero harmonic spinor for each fixed spin structure. The surrounding discussion records positive results in several dimensions and for spheres, while the general case remains unresolved.