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 χp 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 <χp≤2 . Confident measurements of gravitational-wave sources with χp 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 systematics 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.