Spectral scalar curvature and Urysohn width in dimension three

Every connected complete smooth Riemannian three-manifold without boundary satisfying $-4\Delta +\mathrm{Scal}\ge \lambda >0$ as a quadratic-form inequality admits a continuous map to a graph whose entire fibers have diameter at most $500/\sqrt{\lambda }$ in the original metric. No orientability, spin, compactness, or bounded-geometry assumption is required.