Ends of shrinking gradient ρ-Einstein solitons

We prove that all ends of a gradient shrinking ρ-Einstein soliton are φ-non-parabolic, provided ρ is nonnegative and the soliton has bounded and nonnegative scalar curvature, where the weight φ is a negative multiple of the potential function. We also show these solitons are connected at infinity for $\rho \in [0,1/2(n-1\left)\right)$, $n\ge 4$, and assuming a suitable bound for the scalar curvature.