Homogeneity conjecture for projectively induced Kähler–Einstein manifolds

Let $(M,g)$ be a complete Kähler–Einstein manifold, and let $\mathbb{C}\mathrm{P}^N$ be complex projective space, where $N$ may be finite or infinite. A Kähler immersion is an isometric holomorphic immersion. Homogeneity conjecture. If $(M,g)$ admits a Kähler immersion into $\mathbb{C}\mathrm{P}^N$, then $(M,g)$ is homogeneous. The source states that the only known projectively induced Kähler–Einstein manifolds are homogeneous, while also noting that the conjecture fails when the ambient space is $\mathbb{C}\mathrm{P}^{\infty}$; thus the stated conjecture is refuted in that setting.