Umehara's exception conjecture for Kähler immersions into complex hyperbolic space and Hilbert space

Let $(M,g)$ be a Kähler–Einstein manifold. A Kähler immersion is an isometric holomorphic immersion into a Kähler manifold. The spaces $\mathbb{C}\mathrm{H}^{\infty}$ and $l^2(\mathbb{C})$ denote infinite-dimensional complex hyperbolic space and complex Hilbert space, respectively. Umehara's exception conjecture. If $(M,g)$ admits a Kähler immersion into $\mathbb{C}\mathrm{H}^{\infty}$ or $l^2(\mathbb{C})$, then either $(M,g)$ is totally geodesic or

$$(M,g)=(\mathbb{C}\mathrm{H}^{n_1}\times\cdots\times\mathbb{C}\mathrm{H}^{n_r},c_1g_{hyp}\oplus\cdots\oplus c_rg_{hyp})$$

for positive constants $c_1,\dots,c_r$ and some $r\in\mathbb{N}$. This conjecture proposes that the product of scaled complex hyperbolic spaces is the only exception to the relevant extension of Umehara's result, apart from the totally geodesic case; its resolution is not specified in the source.