The nonarithmetic lattice existence conjecture for complex hyperbolic groups

Let $n$ be a positive integer. Nonarithmetic lattice existence conjecture. For each $n$, the group $\operatorname{PU}(n,1)$ contains a nonarithmetic lattice. Nonarithmetic lattices are known in $\operatorname{PU}(2,1)$ and $\operatorname{PU}(3,1)$, while the source emphasizes that existing construction techniques become increasingly difficult as $n$ grows; the assertion for every $n$ remains open.