The generic-metric conjecture for harmonic spinors

Let $M$ be a compact spin manifold and let $D_g$ be its Dirac operator for a Riemannian metric $g$ on $M$. The Atiyah–Singer index theorem imposes a lower bound on the dimension of $\ker D_g$, and harmonic spinors are sections in this kernel.

Generic-metric conjecture. For a generic Riemannian metric on any compact spin manifold, the space of harmonic spinors is no larger than forced by the index theorem.

This predicts that the topological lower bound for the multiplicity of the zero eigenvalue is sharp for generic metrics. The source states that it is known in dimensions at most $4$ and, in dimensions at least $5$, for a large class including all simply connected manifolds; the general case remains open.