Ends of shrinking gradient $\rho$-Einstein solitons

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