Dumitrescu–Zeghib completeness conjecture for constant-curvature holomorphic Riemannian manifolds

Let $M$ be a compact complex manifold endowed with a holomorphic Riemannian metric of constant non-vanishing curvature. The dimension of $M$ is the complex dimension. Dumitrescu–Zeghib conjecture. The manifold $M$ is complete. In particular, its dimension is $3$ or $7$. The assertion concerns the unresolved completeness problem for compact complex manifolds with such metrics; the dimensional restriction is included as part of the stated claim.