Did Sgr cause the vertical waves in the solar neighbourhood?

The vertical distribution of stars in the solar neighbourhood is not in equilibrium but contains a wave signature in both density and velocity space originating from a perturbation. With the discovery of the phase-space spiral in Gaia data release 2, determining the origin of this perturbation has become even more urgent. We develop and test a fast method for calculating the perturbation from a passing satellite on the vertical component of a part of a disc galaxy. This fast method allows us to test a large variety of possible perturbations to the vertical disc very quickly. We apply our method to the range of possible perturbations to the solar neighbourhood stemming from the recent passage of the Sagittarius dwarf galaxy (Sgr), varying its mass, mass profile, and present-day position within their observational uncertainties, and its orbit within different realistic models for the Milky Way's gravitational potential. We find that we are unable to reproduce the observed asymmetry in the vertical number counts and its concomitant breathing mode in velocity space for any plausible combination of Sgr and Milky-Way properties. In all cases, either the amplitude or the perturbation wavelength of the number-count asymmetry and of the oscillations in the mean vertical velocity produced by the passage of Sgr is in large disagreement with the observations from Gaia DR2. We conclude that Sgr cannot have caused the observed oscillations in the vertical disc or the Gaia phase-space spiral.