Uniform log Yau–Tian–Donaldson conjecture for cscK cone metrics

Let $X$ be a Kähler manifold, let $D$ be a smooth effective divisor on $X$, let $L$ be an ample line bundle, and let $((X,L);D)$ be the associated polarised pair. A cscK cone metric is a constant scalar curvature Kähler metric with cone singularities along $D$.

Uniform log Yau–Tian–Donaldson conjecture. The polarised pair $((X,L);D)$ admits a cscK cone metric if and only if it is uniformly log $K$-stable.

This is the expected uniform-stability refinement of the logarithmic Yau–Tian–Donaldson correspondence. The paper proves that existence implies $G$-uniform log $K$-stability for the automorphism group preserving $D$ in the stated setting, but does not establish the converse in general.