The folklore conjecture on projectively induced Kähler–Einstein metrics

Let $g$ be a projectively induced Kähler–Einstein metric on a compact manifold $X$, meaning that there is an immersion $\varphi\colon X\to \mathbb{P}^s$ into a complex projective space with $g=\varphi^*g_{\mathrm{FS}}$, where $g_{\mathrm{FS}}$ is the Fubini–Study metric.

Folklore conjecture. Then $(X,g)$ is a homogeneous space.

The conjecture proposes that projectively induced Kähler–Einstein metrics occur only on homogeneous spaces. The introduction notes that all previously known examples of Kähler–Einstein submanifolds of projective space were homogeneous, while the paper studies toric Fano examples that provide evidence for this broader characterization.