The conjecture that compact rationally connected Kähler manifolds are Oka-1

A complex manifold $X$ is rationally connected if every pair of points in $X$ can be connected by a rational curve $\mathbb{CP}^1\to X$. A compact Kähler manifold is a compact complex manifold admitting a Kähler metric, and an Oka-1 manifold is a complex manifold with the Oka-1 property studied in the paper. The rationally connected Kähler Oka-1 conjecture. Every compact rationally connected Kähler manifold is an Oka-1 manifold. This is the explicit conjecture proposed in the paper’s section on rationally connected manifolds. It extends the paper’s results on Oka-1 manifolds to compact rationally connected Kähler manifolds, and no proof or disproof is supplied here.