Minimal Euler characteristic conjecture for compact 3-manifold groups

Let $M$ be a compact $3$-manifold with prime decomposition

$$M=(\#_{i=1}^mM_i)\#(\#_{j=1}^n N_j)\#(\#_{l=1}^p Q_l)\#(\#_{e=1}^q S_e),$$

where the prime factors are as in (1.1) and categories (i)–(iv): the $M_i$ are closed prime $3$-manifolds with infinite fundamental group that are not $S^2$- or $RP^2$-bundles over $S^1$, the $N_j$ are closed prime $3$-manifolds with finite fundamental group, the $Q_l$ are prime $3$-manifolds with nonempty boundary, and the $S_e$ are $S^2$- or $RP^2$-bundles over $S^1$. Let $\pi=\pi_1(M)$, and write $\chi_4(\pi)$ for the minimal Euler characteristic of a closed $4$-manifold with fundamental group $\pi$ and $p(\pi)$ for the corresponding group invariant. Minimal Euler characteristic conjecture.

$$\chi_4(\pi)=p(\pi)=2-2(p+q)+\chi(\partial M).$$

This conjecture proposes a formula for the minimal Euler characteristic and the invariant $p(\pi)$ of fundamental groups of compact $3$-manifolds, extending known results for important classes of closed aspherical $3$-manifold groups. Its general status is not established in the supplied text.