Are All Rational Manifolds Algebraically Elliptic?

Determine whether every rational complex manifold is algebraically elliptic.

Background

While Corollary 3.5 proves that compact rational manifolds are algebraically Oka‑1, the stronger algebraic ellipticity (which implies several powerful algebraic Oka properties) is not known for all rational manifolds.

Establishing algebraic ellipticity for rational manifolds would significantly enhance the toolkit for constructing and approximating holomorphic maps from algebraic sources and would deepen the relationship between rationality and holomorphic flexibility.