The theory of hole superconductivity predicts that in superconductors the charged superfluid is about a million times more rigid than the normal electron fluid. We point out that this physics should give rise to large changes in the bulk and surface plasmon dispersion relations of metals entering the superconducting state, that have not yet been experimentally detected and would be in stark contradiction with the expected behavior within conventional BCS-London theory. We also propose that this explains the puzzling experimental observations of Avramenko et al  on electron sound propagation in superconductors and the puzzling experiments of W. de Heer et al  detecting large electric dipole moments in small metal clusters, as well as the Tao effect  on aggregation of superconducting microparticles in an electric field. Associated with the enhanced charge rigidity is a large increase in the electric screening length of superconductors at low temperatures that has not yet been experimentally detected. The physical origin of the enhanced charge rigidity and its relation to other aspects of the theory of hole superconductivity is discussed.