Properly $3$-realizable groups

A finitely presented group $G$ is said to be properly $3$-realizable if there exists a compact $2$-polyhedron $K$ with $\pi_1(K) \cong G$ and whose universal cover $\tilde{K}$ has the proper homotopy type of a (p.l.) $3$-manifold with boundary. In this paper we show that, after taking wedge with a $2$-sphere, this property does not depend on the choice of the compact $2$-polyhedron $K$ with $\pi_1(K) \cong G$. We also show that (i) all $0$-ended and $2$-ended groups are properly $3$-realizable, and (ii) the class of properly $3$-realizable groups is closed under amalgamated free products (HNN-extensions) over a finite cyclic group (as a step towards proving that $\infty$-ended groups are properly $3$-realizable, assuming $1$-ended groups are).