The anti-canonical polar cylinder criterion for K-polystability of Fano varieties

Let $X$ be a Fano variety with at most Kawamata log terminal singularities. An anti-canonical polar cylinder means a cylinder polarized by $-K_X$. The anti-canonical cylinder conjecture. If $X$ does not contain $(-K_X)$-polar cylinders, then $X$ is $K$-polystable. This conjecture proposes that the absence of anti-canonical polar cylinders gives a sufficient condition for $K$-polystability, complementing the role of cylinders as potential sources of destabilizing test configurations. Its general status is unresolved.