Does the quadrupole moment of supermassive black holes equal $-J^{2}/(M c^{2})$?
In plain words
A Kerr black hole's shape is fixed by its mass and spin, so its flattening, measured by the quadrupole moment, must take one exact value. Small bodies spiraling into massive black holes, to be observed by the LISA space detector, can measure it.
Precise statement
For Kerr the mass quadrupole moment is $Q=-J^{2}/(M c^{2})$ (in g cm^2). Using extreme mass-ratio inspirals (a stellar-mass body orbiting a supermassive black hole) observed by LISA over about $10^{5}$ orbital cycles, measure M, J and Q independently and test this relation; Barack and Cutler (2007) estimated fractional precision of about $10^{-4}\ \text{to}\ 10^{-2}$. An answer is the measured deviation of $Q$ from $-J^{2}/(M c^{2})$ with its uncertainty.
What would settle it
LISA extreme mass-ratio inspiral observations with an independent fit of the central quadrupole moment.
Status in the literature
Unverified note
LISA is planned for launch in the mid-2030s.