  We study the connection between the fundamental groups of complex hyperbolic
manifolds and those of spaces arising from the (relative) strict
hyperbolization process due to Charney--Davis and
Davis--Januszkiewicz--Weinberger. Viewing a non-uniform lattice $\Gamma$ in
$\text{PU}(n,1)$ as a relatively hyperbolic group with respect to its cusp
subgroups in the usual way, we show that when $n\geq 2$, $\Gamma$ is not
isomorphic to any relatively hyperbolic group arising from the relative strict
hyperbolization process, via work of Lafont--Ruffoni. We also prove that a
uniform lattice in $\text{PU}(n,1)$ is not the fundamental group of a
Charney-Davis strict hyperbolization when $n\geq 2$, assuming the initial
complex satisfies some mild conditions.
