Weak Besse conjecture on rigidly scalar-flat manifolds

Let $M$ be a compact, locally irreducible Riemannian manifold that is rigidly scalar-flat, meaning that it admits a scalar-flat metric but no metric of positive scalar curvature. Its holonomy is called special if it is not generic in the sense of the source, namely if its restricted holonomy group is neither $SO(n)$ nor, in the locally Kählerian case, $U(n/2)$. Weak Besse conjecture. Every locally irreducible rigidly scalar-flat compact manifold has special holonomy. This conjecture expresses the proposed reason that such manifolds are Ricci-flat: rigidity should arise from special holonomy. The source gives no resolution, so its status remains open.