Properness criterion for the reduced alpha-K-energy

Prove or disprove that, on a compact Kähler manifold with discrete automorphism group, the reduced alpha-K-energy is proper if and only if a Kähler–Yang–Mills metric exists.

Background

The paper proves that existence of a Kähler–Yang–Mills metric implies that both the reduced alpha-K-energy and the full alpha-K-energy are bounded from below. It then places this one-sided result in the broader moment-map and Kempf–Ness framework, where properness of an energy functional is expected to characterize the existence of a canonical metric.

The authors explicitly formulate the converse-and-equivalence assertion as a conjecture. The proposed criterion would strengthen boundedness from below to properness and provide an existence characterization for Kähler–Yang–Mills metrics under the discrete-automorphism hypothesis.