Sun–Zhang's K-polystability conjecture for polarized Fano fibrations

A polarized Fano fibration is a triple $(\pi:X\to Y,\xi)$ consisting of a fibration from a quasiprojective variety $X$ to an affine variety $Y$, with $-K_X$ $[?]$-ample, together with the relevant vector field $\xi$. A Kähler–Ricci shrinker is a metric $(\omega,\xi)$ satisfying the shrinking Kähler–Ricci soliton equation.

Sun–Zhang's conjecture. A polarized Fano fibration $(\pi:X\to Y,\xi)$ admits a Kähler–Ricci shrinker $(\omega,\xi)$, unique up to automorphisms of $X$ preserving $\xi$, if and only if it is K-polystable.

This is a Yau–Tian–Donaldson-type conjecture relating the existence and uniqueness of Kähler–Ricci shrinkers to algebraic K-polystability. The paper proves the existence direction under the additional assumption that the Ricci curvature of the shrinker decays at infinity, while the full equivalence remains open.