Conjecture: Rationally Connected Projective Manifolds Are Oka-1

Prove that every rationally connected projective complex manifold is an Oka-1 manifold, thereby establishing the Oka-1 property across the entire class of rationally connected projective manifolds.

Background

Proposition 1.7 shows that every projective algebraically Oka-1 manifold is rationally connected, linking Oka-type properties to rational connectedness. The authors obtain positive results for several subclasses (e.g., rational and algebraically elliptic manifolds) and birational invariance statements, but the full implication from rational connectedness to the Oka-1 property remains unsettled.

They note that this conjecture would follow from a theorem of Gournay, but the proof details are not clear to them. Establishing this conjecture would unify several strands connecting complex geometry, rational connectedness, and Oka-type approximation properties.