Uniqueness conjecture for high-temperature μ-cscK metrics

Let $(X,L)$ be a polarized variety, and let $\mu^\lambda$-cscK metrics be the metrics satisfying the $\mu^\lambda$-constant scalar curvature Kähler equation for a real parameter $\lambda$. Two metrics are equivalent modulo the automorphism group if they lie in the same orbit under its action.

High-temperature uniqueness conjecture. For $\lambda\leq 0$, $\mu^\lambda$-cscK metrics are unique modulo the automorphism group.

This conjectures uniqueness in the high-temperature regime. The surrounding discussion states that uniqueness can fail when $-\lambda\ll 0$, while the claim for $\lambda\leq 0$ remains open in the supplied text.