Let $M$ be a compact 4-manifold with finite fundamental group such that the prime factorization of its boundary $\partial M$ is connected and has no aspherical factors. Assuming $\partial M$ is incompressible in $M$, then $\partial M$ is spherical by the elliptization theorem. Barring making this assumption, are there other conditions that guarantee $\partial M$ is spherical?

Let $M$ be a compact 4-manifold with finite fundamental group such that the prime factorization of its boundary $\partial M$ is connected and has no aspherical factors. Assuming $\partial M$ is incompressible in $M$, then $\partial M$ is elliptic by the elliptization theorem. Barring making this assumption, are there other conditions that guarantee $\partial M$ is elliptic? By elliptic I mean a spherical space form. [I've edited this question because I earlier mistakenly used the terminology spherical.]

Let $M$ be a compact 4-manifold with finite fundamental group such that the prime factorization of its boundary $\partial M$ has no aspherical factors. Assuming $\partial M$ is incompressible in $M$, then $\partial M$ is spherical by the elliptization theorem. Barring making this assumption, are there other conditions that guarantee $\partial M$ is spherical?

Let $M$ be a compact 4-manifold with finite fundamental group such that the prime factorization of its boundary $\partial M$ is connected and has no aspherical factors. Assuming $\partial M$ is incompressible in $M$, then $\partial M$ is spherical by the elliptization theorem. Barring making this assumption, are there other conditions that guarantee $\partial M$ is spherical?