The one-ended non-negative Ricci curvature conjecture for the Riesz transform

Let $M$ be a Riemannian manifold satisfying a Sobolev inequality, with non-negative Ricci curvature outside a compact set and only one end. The one-ended Riesz transform conjecture. The Riesz transform on $M$ should be bounded on $L^p$ for every $1<p<\infty$.

The conjecture concerns extending the known perturbation results for the Riesz transform to manifolds whose topology and metric are changed on a compact set without introducing additional ends. The surrounding discussion indicates that the stronger assumption of no non-zero $L^2$ harmonic $1$-forms is expected to be unnecessary, but does not establish the conjecture.