Székelyhidi's relative K-polystability conjecture for extremal Kähler metrics

Let $X$ be a polarized manifold with polarization class $[\theta]$, and let $T$ be a maximal torus of automorphisms of $X$. The manifold is relatively K-polystable when it is K-stable relative to $T$.

Székelyhidi's conjecture. $X$ admits an extremal Kähler metric in the class of the polarization if and only if it is relatively K-polystable.

This is the Yau–Tian–Donaldson conjecture for extremal Kähler metrics. The implication from existence of an extremal Kähler metric to relative K-polystability is known, while the converse remains open.