Is $M=E_{7(7)}/SU(7)\times\mathbb{R}^{+}$ a (pseudo)Kähler-Hodge manifold? Open problem

Question

I have been told that the following is an open problem in mathematics, but I am pretty sure that experts in the topic surely know the answer.

Is the manifold

$$M=\frac{E_{7(7)}}{SU(7)}\times \mathbb{R}^{+}$$

a (pseudo)Kähler-Hodge manifold? Just to be clear, by this I mean if $M$ can be equipped with a (pseudo)Kähler structure, namely a complex structure $J$, a symplectic structure $\omega$ and a compatible metric $g$ (of indefinite signature) such that $\omega$ is integral, that is, $[\omega] \in H^{2}(M,\mathbb{Z})$, together with a hermitian line-bundle $\mathcal{L}\to M$ such that its first Chern class is

$c_{1}(\mathcal{L}) = [\omega]$

$E_{7(7)}$ refers to the split real form of $E_{7}$.

Thanks.

Edit: I have been required to provide the reference that states this as an open problem. Here it is:

http://arxiv.org/abs/0804.1362

(see section 3.3) That reference also explains the relevance of this problem in describing the moduli space of M-theory flux compactifications on seven-dimensional manifolds.

Answer 1

If i am not wrong the following is true, G compact connected Lie group, H the centralizer of a torus in G, then G/H is a projective manifold, does this answer your question??