Existence of Kähler–Einstein metrics on the manifolds $\mathcal M_{s,p,n}$

Let $\mathcal M_{s,p,n}$ be the manifold considered above, and let $s,p,n$ be its defining parameters. A line bundle $L$ on a projective variety $X$ is called big when its highest self-intersection number $(L^n)$ is positive and nef when $(L\cdot C)\geq 0$ for every curve $C$ on $X$. Existence conjecture. There are Kähler–Einstein metrics on the manifolds $\mathcal M_{s,p,n}$. This conjecture extends the established existence results for $\mathcal M_{p,p,2p}$ when $p\leq 5$ and for $\mathcal M_{s,p,n}$ when $r\leq 2$, while the general case remains open.