Is the g-factor of a charged black hole 2 or is it 2+alpha/pi?

Question

Many papers over the past 40 years discuss the magnetic moment and the $g$-factor of a black hole in the zeroth approximation and get the result $g=2$.

But what happens to the next approximation? Is the precise value

$g=2$  or $g= 2 + \alpha/\pi$ ?

Here $\alpha$ is the fine structure constant. Are there papers (I did not find any) or ideas on the issue? Will these values ever be tested in measurements?

Comments

A black hole, as a macroscopic body, may have any $g$-factor, in my humble opinion. It really depends on particulat properties involved.

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All papers (dozens) state that g=2 for black holes. But none mentions corrections in alpha.