Kobayashi's complex sphere conjecture

Let $n$ be a positive integer, and let $SO(n+1,\mathbb{C})/SO(n,\mathbb{C})$ be the $n$-dimensional complex sphere. Kobayashi's complex sphere conjecture. The complex sphere admits a compact Clifford–Klein form if and only if $n=1,3$ or $7$. The cases $n=1,3,7$ are known to admit such forms; the conjectural assertion is the nonexistence in all other dimensions.