Equivariant characterization of K-polystability

Prove or disprove that, for every polarized projective variety $(Y,H)$ and every reductive subgroup $G\subset\operatorname{Aut}(Y,H)$, nonnegativity of the Donaldson–Futaki invariant on every $G$-equivariant test configuration, with equality only when the normalization is a product, is equivalent to K-polystability.

Background

The paper records a conjecture of Codogni and Stoppa concerning the reduction of K-polystability to equivariant test configurations. The conjecture requires both nonnegativity of the Donaldson–Futaki invariant and a precise characterization of the zero-invariant case.

The paper proves this conjectural equivalence only for the particular fivefold constructed in its main theorem, not for arbitrary polarized varieties or reductive subgroups.

References

We remark that Codogni and Stoppa conjectured that, for every reductive subgroup $G\subset\Aut(Y,H)$, K-polystability is equivalent to nonnegativity of the Donaldson--Futaki invariant on every $G$-equivariant test configuration, with equality if and only if its normalization is a product.

— Disproof of the Yau--Tian--Donaldson conjecture  (2608.19301 - Liu, 19 Aug 2026) in Remark 1.2, Section 1, “Background”