Bruce Hunt's hybrid construction conjecture for nonarithmetic lattices

For every $n\ge 2$, consider arithmetic lattices $\Gamma_1,\Gamma_2<\operatorname{SU}(n-1,1)$ and $\Gamma_3<\operatorname{SU}(n-2,1)$. Suppose $\Gamma_3$ is isomorphic to subgroups of both $\Gamma_1$ and $\Gamma_2$, and form the amalgamated product $\Gamma_0=\Gamma_1\star_{\Gamma_3}\Gamma_2$. Bruce Hunt's hybrid construction conjecture. There exist such lattices and an epimorphism $\rho:\Gamma_0\to\Gamma<\operatorname{SU}(n,1)$ that is injective on $\Gamma_1$ and $\Gamma_2$, with image a nonarithmetic lattice $\Gamma<\operatorname{SU}(n,1)$. By analogy with the Gromov--Piatetski-Shapiro construction, this proposes a hybrid construction in complex hyperbolic geometry; the source notes that $\rho$ cannot be injective on all of $\Gamma_0$.