Finiteness conjecture for essential projective transformations

Let $(M,g)$ be a compact pseudo-Riemannian manifold. Write ${\sf Proj}(M,g)$ for its projective transformation group and ${\sf Aff}(M,g)$ for its affine transformation group.

Finiteness conjecture. Unless $(M,g)$ is a finite quotient of the standard Riemannian sphere,

$${\sf Proj}(M,g)/{\sf Aff}(M,g)$$

is finite. The analogous question is posed with compactness replaced by completeness; this does not contain the compact case because compact pseudo-Riemannian manifolds need not be complete. The conjecture concerns rigidity of the projective group and the characterization of pseudo-Riemannian manifolds admitting essential projective transformations.