Macroscopic Schoen conjecture for manifolds with non-zero simplicial volume

We prove that given a hyperbolic manifold endowed with an auxiliary Riemannian metric whose sectional curvature is negative and whose volume is sufficiently small in comparison to the hyperbolic one, we can always find for any radius at least $1$ a ball in its universal cover whose volume is bigger than the hyperbolic one. This result is deduced from a non-sharp macroscopic version of a conjecture by R. Schoen about scalar curvature, whose proof is a variation of an argument due to M. Gromov and based on a smoothing technique. We take the opportunity of this work to present a full account of this technique which involves simplicial volume and deserves to be better known.