Spin precession in merging black-hole binaries is a treasure trove for both
astrophysics and fundamental physics. There are now well-established strategies
to infer from gravitational-wave data whether at least one of the two black
holes is precessing. In this paper we tackle the next-in-line target, namely
the statistical assessment that the observed system has two precessing spins.
We find that the recently-developed generalization of the effective precession
spin parameter ${\chi}_{\mathrm{p}}$ is a well-suited estimator to this task. With
this estimator, the occurrence of two precessing spins is a necessary (though
not sufficient) condition to obtain values $1<{\chi}_{\mathrm{p}}\le 2$ . Confident
measurements of gravitational-wave sources with ${\chi}_{\mathrm{p}}$ values in
this range can be taken as a conservative assessment that the binary presents
two precessing spins. We investigate this argument using a large set of >100
software injections assuming anticipated LIGO/Virgo sensitivities for the
upcoming fourth observing run, O4. Our results are very encouraging, suggesting
that, if such binaries exist in nature and merge at a sufficient rate, current
interferometers are likely to deliver the first confident detection of merging
black holes with two precessing spins. We investigate prior effects and
waveform systemics and, though these need to be better investigated, did not
find any confident false-positive case among all the configurations we tested.
Our assessment should thus be taken as conservative.