Tangent-flow-at-infinity conjecture for ancient Kähler-Ricci flows

Let $(X,\omega(t))$, $t\in(-\infty,0]$, be a non-collapsed ancient Kähler-Ricci flow. A non-collapsing weak Fano fibration is a fibration $\pi:X\rightarrow Y$ with $X$ klt, $Y$ normal affine, and $-K_X$ relatively nef and big. A negative valuation is a function $d$ on the nonzero elements of the global anti-canonical ring satisfying

$$d(s_1+s_2)\geq\min\{d(s_1),d(s_2)\},\qquad d(s_1s_2)=d(s_1)+d(s_2).$$

It is K-polystable when its associated graded algebra is finitely generated and, with reversed grading, defines a K-polystable polarized Fano fibration.

Tangent flows at infinity conjecture. Given a non-collapsed ancient Kähler-Ricci flow $(X,\omega(t))$, $t\in(-\infty,0]$:

1. $X$ admits a non-collapsing weak Fano fibration structure and the flow naturally induces a K-polystable negative valuation.
2. The tangent flow at infinity of $\omega(t)$ is unique and is given by the Kähler-Ricci shrinker determined by the K-polystable negative valuation via the K-polystability criterion for Kähler-Ricci shrinkers.

This is a global analogue of the weighted-volume degeneration conjecture and predicts uniqueness and algebraic characterization of tangent flows at infinity. It remains open.