Hermitian Einstein characterization of absolute Weyl-functional minimizers

Let $M$ be a smooth compact $4$-manifold with an Einstein metric $g$ of positive Einstein constant, and suppose that $g$ is Hermitian with respect to an integrable complex structure $J$ on $M$. Let $[g]$ denote its conformal class and let $\mathscr{W}$ be the Weyl functional.

Hermitian Einstein minimizer conjecture. The conformal class $[g]$ is an absolute minimizer of $\mathscr{W}$. Moreover, for the given $M$, every absolute minimizer arises from a metric satisfying these conditions.

The conjecture is presented as evidence-based speculation in the paper's Problems and Prospects section. It strengthens the preceding interpretation by proposing both sufficiency and necessity for absolute minimizers.