Fang's extremal metric conjecture for Dolbeault determinants on the sphere

Let $(S^{2},\omega_{0})$ be the standard round sphere, and let $L$ be an ample holomorphic line bundle over $S^{2}$. Consider the determinant of the Dolbeault Laplacian as a functional of smooth Hermitian metrics on $L$.

Fang's conjecture. The upper bound is achieved precisely by the Fubini–Study metric on $L$, up to scaling.

This refines the boundedness conjecture by identifying all maximizers in the spherical ample case. The source presents it as a more precise conjectural form of the boundedness statement.