A 3-manifold $M$ is irreducible if every embedded 2-sphere bounds a 3-ball. Thanks to Papakyriakopoulos's sphere theorem, irreducibility is the same as having $\pi_2(M)=0$. Does irreduciblity imply that the manifold is in fact aspherical, i.e. that $\pi_k(M)=0$ for all $k \geq 2$?

(Or maybe I should say that the universal cover $\tilde M$ is aspherical, but the question is the same in terms of homotopy groups.)

I'd expect that the answer might be yes based on all of the tools we have about geometric classification of 3-manifolds, but I'm not enough of an expert to make those arguments myself. Does someone here have a quick answer?

A 3-manifold $M$ is irreducible if every embedded 2-sphere bounds a 3-ball. Thanks to Papakyriakopoulos's sphere theorem, irreducibility is the same as having $\pi_2(M)=0$. Does irreduciblity imply that the manifold is in fact aspherical, i.e. that $\pi_k(M)=0$ for all $k \geq 2$?

(Or maybe I should say that the universal cover $\tilde M$ is aspherical, but the question is the same in terms of homotopy groups.)

As pointed out by @Matt Zaremsky in the comments, there is an obvious counterexample in $S^3$. But perhaps this is the only counterexample, or the counterexamples are easy to classify?

Given all of the tools we have about geometric classification of 3-manifolds, I expect someone would have a quick answer. I'm just not enough of an expert to make those arguments myself.