The high-dimensional non-reflection lattice conjecture

There exists an integer $N$ such that for every $n\ge N$ and every lattice $\Gamma<\operatorname{PU}(n,1)$, the following hold: $\Gamma$ is not a reflection subgroup, and the underlying space of the orbifold $M_\Gamma$ is not a rational algebraic variety. More ambitiously, that underlying space is a variety of general type. High-dimensional non-reflection lattice conjecture. Such an $N$ exists and these assertions hold for all $n\ge N$. The conjecture is motivated by the known nonarithmetic examples in dimensions $2$ and $3$, which are commensurable with complex reflection subgroups and have rational projective underlying spaces; the proposed obstruction concerns sufficiently high dimensions and remains open.