Shrinking gradient Kähler-Ricci solitons (Kähler-Ricci shrinkers) are fundamental objects in the study of the Kähler-Ricci flow, characterizing much of the behavior of finite-time singularities. Recently, Sun--Zhang have developed an algebraic theory for Kähler-Ricci shrinkers, which in particular implies that such spaces are naturally quasiprojective varieties. Moreover, they propose a YTD correspondence between the existence of such a metric and an algebro-geometric notion of K-stability, analogous to and in fact extending the well-known situations for Fano manifolds and Kähler cones. In this talk, I will discuss the proof of one direction of the correspondence, namely that the existence of a Kähler-Ricci shrinker metric implies K-polystability, in the case that the Ricci curvature decays at infinity. This is joint work with Carlos Esparza.