Recently, Sun--Zhang have shown that the manifold underlying any Kähler--Ricci shrinker is a quasiprojective variety, admitting a so called polarized Fano fibration structure. They have furthermore defined a notion of K-stability for polarized Fano fibrations and formulated a YTD-type conjecture. In this talk, I will present a proof of one direction of the conjecture in the asymptotically conical case: If a manifold M admits an asymptotically conical Kähler--Ricci shrinker then M must be K-polystable. This talk is based on joint work with Charles Cifarelli.