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Are there observational candidates for black holes where the spin axis is not aligned in a constant direction? something that would allow to estimate an inertia tensor anisotropy for the black hole
A bit of searching yielded this paper (Full Text) which claims to have observed a change in the direction of the spin axis of a supermassive black hole. So I guess the answer to your question is "yes".
There's also a popular news article if anyone's interested.
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