Independence

The definition of a P$3$ R-group seems to depend on the choice of the $2$ -complex $X$ . In this respect, given a P$3$ R-group $G$ and any other finite $2$ -complex $Y$ with $\pi_1(Y) \cong G$ , the universal cover of the wedge $Y \vee S^2$ is shown to be proper homotopy equivalent to a $3$ -manifold, see [CL05]. Under which conditions is $\tilde{Y}$ proper homotopy equivalent to a $3$ -manifold itself?