Compactness conjecture for locally projectively induced Kähler–Einstein manifolds

Let $(M,g)$ be a complete Kähler–Einstein manifold. A local Kähler immersion is a Kähler immersion defined on neighborhoods, and $\mathbb{C}\mathrm{P}^N$ denotes complex projective space. Compactness conjecture. If $(M,g)$ admits a local Kähler immersion into $\mathbb{C}\mathrm{P}^N$, then $(M,g)$ is compact. This is presented as a weaker conjecture after the homogeneity conjecture, because a homogeneous Kähler manifold admitting a Kähler immersion into a complex projective space is compact; the source does not give a resolution.