  We prove that all ends of a gradient shrinking $\rho$-Einstein soliton are
$\varphi$-non-parabolic, provided $\rho$ is nonnegative and the soliton has
bounded and nonnegative scalar curvature, where the weight $\varphi$ is a
negative multiple of the potential function. We also show these solitons are
connected at infinity for $\rho \in\left[0,1/2(n-1)\right)$, $n\geq4$, and a
suitable bound for the scalar curvature.
