Toricity of rationally connected varieties with int-amplified endomorphisms

Prove or disprove the conjecture that every rationally connected projective manifold admitting an int-amplified endomorphism is toric.

Background

The authors contrast two rigidity phenomena: nef tangent bundles are associated with the Campana–Peternell conjecture, while rationally connected varieties admitting int-amplified endomorphisms are conjectured to be toric. The paper does not resolve this toricity conjecture.

The examples in the paper show that pseudo-effectivity of the tangent bundle alone is too weak to imply Fano type or almost homogeneity, so the toric classification remains a separate unresolved problem in the dynamical setting.