Quadratic norm conjecture for invariant twistor 2-forms

Let $(M,g,J)$ be a 4-dimensional Kähler geometry with a twistor 2-form $\phi^{tw}$ invariant under a Hamiltonian action of a Lie group $G$. Let $e^\mu$ be the norm of $\phi^{tw}$, and let the moment maps correspond to the maximal torus algebra of $G$.

Quadratic norm conjecture. The norm squared

$$e^{2\mu}$$

of the twistor 2-form is at most a quadratic polynomial in these moment maps.

The observation is proved for the toric Kähler surfaces classified in the paper, where the norm squared is at most quadratic in the moment maps. The conjecture proposes that the same property holds for general four-dimensional Kähler geometries with invariant twistor 2-forms, potentially extending the result beyond the toric setting.