$\newcommand{\Q}{\mathbb{Q}}\newcommand{\R}{\mathbb{R}}\DeclareMathOperator\PU{PU}$Let $G$ be a simple algebraic group over $\Q$ such that $G(\R) \simeq \prod_i G_i$, with each $G_i$ being the Lie group $\PU(p_i,q_i)$ for some $p_i,q_i$ such that $p_i \geq q_i$ and $p_i + q_i = n$.
If $\Gamma < G(\Q)$ is a subgroup such that the image of $\Gamma$ in $G_i$ is a non-arithmetic lattice (for some $i$), then is there any known restriction on $p_j, q_j$ for $j \neq i$ (other than the condition that $G_j$ should be noncompact for some $j \neq i$)?
If it helps, one can assume that $\Gamma$ is contained in an arithmetic subgroup of $G(\Q)$.
(By Margulis's theorem one knows that $q_i$ must be $1$, so the question is asking for other known restrictions.)
