The rational-curves density conjecture for terminal Calabi--Yau models

Let $X'$ be a terminal minimal model with torsion canonical bundle. Suppose that $X'$ has no étale cover which is an Abelian variety. The union of rational curves means the union of all rational curves contained in $X'$.

Rational-curves density conjecture. The union of rational curves in $X'$ is Zariski dense in $X'$.

This is a stronger version of the conjecture that torsion-canonical projective manifolds without rational curves are étale quotients of Abelian varieties. The paper notes that it is known for K3 surfaces, while the general terminal Calabi--Yau and hyperkähler cases remain open.