The arbitrary finite Dirac-spectrum prescription conjecture

Let $M$ be a compact spin manifold with Dirac operator $D_g$ for a Riemannian metric $g$. The spinor bundle imposes algebraic constraints on the spectrum, including quaternionic eigenspaces and spectral symmetry, while the Atiyah–Singer index theorem imposes constraints on the multiplicity of the zero eigenvalue.

Arbitrary finite Dirac-spectrum prescription conjecture. Apart from these algebraic constraints and the index-theoretic constraints on the zero eigenvalue, there is a Riemannian metric on any compact spin manifold whose finite part of the Dirac spectrum is arbitrarily prescribed.

The result preceding this conjecture establishes prescription of finitely many nonzero eigenvalues with multiplicity one after choosing a metric with minimal possible kernel, assuming the generic-metric conjecture. The conjecture asks for arbitrary finite multiplicities subject only to the stated spinorial and index-theoretic restrictions, and remains open in the source.