Birational superrigidity implies K-stability

Let $X$ be a Fano variety, meaning a projective variety with ample anticanonical divisor. It is birationally superrigid if its birational structure has the corresponding superrigidity property, and it is K-stable in the sense of Tian and Donaldson.

Birational superrigidity conjecture. A birationally superrigid Fano variety is K-stable.

K-stability is an algebraic condition closely related to the existence of Kähler–Einstein metrics, while birational superrigidity originates in the birational classification and rationality problem for Fano varieties. The conjecture proposes a connection between these two notions through their shared relationship with singularities of anticanonical divisors and movable linear systems.