The rational-curves conjecture for torsion-canonical manifolds

Let $X$ be a projective manifold with torsion canonical bundle and no rational curves. An étale quotient of an Abelian variety is a variety obtained as a quotient of an Abelian variety by a finite étale action.

Rational-curves conjecture for torsion-canonical manifolds. The manifold $X$ is an étale quotient of an Abelian variety.

By the Bogomolov--Beauville decomposition theorem, this is equivalent to asserting that Calabi--Yau and hyperkähler manifolds admit rational curves. The conjecture is known in several cases, including K3 surfaces and some Calabi--Yau threefolds, but remains open in general.