  Recently, strong evidence was found for the presence of higher-order modes in
the gravitational wave signals GW190412 and GW190814, which originated from
compact binary coalescences with significantly asymmetric component masses.
This has opened up the possibility of new tests of general relativity by
looking at the way in which the higher-order modes are related to the basic
signal. Here we further develop a test which assesses whether the amplitudes of
sub-dominant harmonics are consistent with what is predicted by general
relativity. To this end we incorporate a state-of-the-art waveform model with
higher-order modes and precessing spins into a Bayesian parameter estimation
and model selection framework. The analysis methodology is tested extensively
through simulations. We investigate to what extent deviations in the relative
amplitudes of the harmonics will be measurable depending on the properties of
the source, and we map out correlations between our testing parameters and the
inclination of the source with respect to the observer. Finally, we apply the
test to GW190412 and GW190814, finding no evidence for violations of general
relativity.
