We show scalar-mean curvature rigidity of warped products of round spheres of dimension at least 2 over compact intervals equipped with strictly log-concave warping functions. This generalizes earlier results of Cecchini-Zeidler to all dimensions.
Moreover, we show scalar curvature rigidity of round spheres of dimension at least 3 minus two antipodal points, thus resolving a problem in Gromov's ``Four Lectures'' in all dimensions.
Our arguments are based on spin geometry.
