  We provide an extension of the analysis on whether an extended test body can
follow a geodesic trajectory given by Mukherjee, S., Lukes-Gerakopoulos, G. and
Nayak, R. K. (2022), Extended bodies moving on geodesic trajectories, General
Relativity and Gravitation, 54(9), 113, arXiv: 1907.05659. In particular, we
consider a test body in a pole-dipole-quadrupole approximation under the
Ohashi-Kyrian-Semer\'{a}k spin supplementary condition moving in the
Schwarzschild and Kerr background. Using orbital setups under which a
pole-dipole body can follow geodesic motion, we explore under which conditions
this can take place also in the pole-dipole-quadrupole approximation, when only
the mass quadrupole is taken into account. For our analysis we employ the
assumption that the dipole contribution and the quadrupole contribution vanish
independently.
