  Supermassive binaries detectable by the future space gravitational-wave
interferometer LISA might allow to distinguish black holes from ultracompact
horizonless objects, even when the latter are motivated by quantum-gravity
considerations. We show that a measurement of very small tidal Love numbers at
the level of $10\%$ accuracy (as achievable with "golden binaries") may also
allow to distinguish between different models of these exotic compact objects,
even when taking into account an intrinsic uncertainty in the object radius
putatively due to quantum mechanics. We argue that there is no conceptual
obstacle in performing these measurements, the main challenge remains the
detectability of small tidal effects and an accurate waveform modelling. Our
analysis uses only coordinate-independent quantities related to the proper
radial distance and the total mass of the object.

General Relativity and Quantum Cosmology;High Energy Astrophysical Phenomena;High Energy Physics - Phenomenology