On gradient $\rho$-Einstein solitons with Bach tensor radially nonnegative

In this paper, we study $n$-dimensional gradient $\rho$-Einstein solitons whose Bach tensor is radially nonnegative. Under this assumption, we show that such $\rho$-Einstein solitons are locally warped products of an interval and an Einstein manifold, provided either $\rho\neq0$ or $\rho=0$ and the soliton is rectifiable. We obtain as a consequence that these solitons must have harmonic Weyl tensor and vanishing Bach tensor. We also finish the classification of complete locally conformally flat steady $\rho$-Einstein solitons and classify these manifolds when their Bach tensor is radially nonnegative and $n\in\{3,4\}$.