Rigidity conjecture for maximal representations of complex hyperbolic lattices

Let $\Gamma<\operatorname{SU}(n,1)$ be a cocompact complex hyperbolic lattice with $n>1$, and let $G$ be a simple Lie group of Hermitian type. A representation $\rho\colon\Gamma\to G$ is maximal when

$$\tau(\rho)=\operatorname{rk}(G)\operatorname{Vol}(\Gamma\delimiter"526E30F\mathopen{}\mathbb{B}^n).$$

The standard diagonal embedding is

$$\rho_{\mathrm{std}}\colon \Gamma<\operatorname{SU}(n,1)\hookrightarrow\operatorname{SU}(nq,q)\hookrightarrow\operatorname{SU}(p,q).$$

Rigidity conjecture. If $\rho\colon\Gamma\to G$ is maximal, then $G=\operatorname{SU}(p,q)$ with $p\geq nq$, and $\rho$ is a trivial deformation of $\rho_{\mathrm{std}}$, namely

$$\rho(\gamma)=\chi(\gamma)\rho_{\mathrm{std}}(\gamma),$$

where $\chi\colon\Gamma\to Z_G\bigl(\rho_{\mathrm{std}}(\operatorname{SU}(n,1))\bigr)$.

This conjecture predicts that maximal representations of cocompact complex hyperbolic lattices in rank one are highly rigid, with the only freedom coming from centralizer-valued deformations of the standard diagonal embedding. The source presents the assertion as a conjecture; no resolution is supplied here.