The multipole bound for axisymmetric black holes

Let $M$ be the mass, $J$ the angular momentum, and $M_2$ the mass quadrupole of an axisymmetric black hole. Multipole bound conjecture. The quadrupole should satisfy

$$M_2\geq-\frac{J^2}{M}.$$

The bound is saturated by Kerr. The source gives no resolution status for general axisymmetric black holes.