  We investigate the prospect of performing a null test of binary black hole
(BBH) nature using spin-induced quadrupole moment (SIQM) measurements. This is
achieved by constraining a deviation parameter ($\delta\kappa$) related to the
parameter ($\kappa$) that quantifies the degree of deformation due to the spin
of individual binary components on leading (quadrupolar) spin-induced moment.
Throughout the paper, we refer to $\kappa$ as the SIQM parameter and
$\delta\kappa$ as the SIQM-deviation parameter. The test presented here extends
the earlier SIQM-based null tests for BBH nature by employing waveform models
that account for double spin-precession and higher modes. We find that waveform
with double spin-precession gives better constraints for $\delta\kappa$,
compared to waveform with single spin-precession. We also revisit earlier
constraints on the SIQM-deviation parameter for selected GW events observed
through the first three observing runs (O1-O3) of LIGO-Virgo detectors.
Additionally, the effects of higher-order modes on the test are also explored
for a variety of mass-ratio and spin combinations by injecting simulated
signals in zero-noise. Our analyses indicate that binaries with mass-ratio
greater than 3 and significant spin precession may require waveforms that
account for spin-precession and higher modes to perform the parameter
estimation reliably.
