Recently, A. Carlotto and C. Li proved a complete topological classification of those compact, connected and orientable $3$-manifolds with boundary which support Riemannian metrics of positive scalar curvature and mean-convex boundary. Namely, if $M^3$ is such a manifold, then there exist integers $A, B, C, D \geq 0$ such that $M$ is diffeomorphic to a connected sum of the form \begin{align*} P_{\gamma_1} \# \cdots \# P_{\gamma_A} \# \mathbb{S}^3/ {\Gamma_1} \# \cdots \# \mathbb{S}^3 / {\Gamma_B} \# \left( \#_{i=1}^C \mathbb{S}^2 \times \mathbb{S}^1 \right) \setminus \left( \sqcup_{i=1}^D B_i^3 \right), \end{align*} where $P_{\gamma_i}$, $i \leq A$, are genus $\gamma_i$ handebodies; $\Gamma_i$, $i \leq B$, are finite subgroups of $SO(4)$ acting freely on $\mathbb{S}^3$, $B_i^3$, $i \leq D$, are disjoint $3$-balls in the interior.

**My question:** Can we classify, in terms of $A,B, C, D$, the $3$-manifolds $M$ of the form above in which any smoothly embedded $2$-sphere in the interior separates $M$?

For instance, if $(A,B,C,D) = (1, 0, 0, 0)$, then this holds. Indeed, if $M = P_{\gamma_1}$, then $H_2(M) = 0$, so the connecting homomorphism $H_2(M, \partial M;\mathbb{Z}) \to H_1(\partial M; \mathbb{Z})$ is injective. Since an embedded $2$-sphere has no boundary, it lies in the kernel of this map, and thus equals to $0$ in $H_2(M, \partial M; \mathbb{Z})$. This means it separates $M$.