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. Completeness conjecture. The metric on $M$ is complete; in particular, $M$ has dimension $3$ or $7$. Examples of such manifolds are known in dimension $3$, while completeness is explicitly identified in the source as an open question; the dimensional conclusion is tied to the conjectural completeness assertion.