The lattice-preservation conjecture for convex co-compact complex hyperbolic representations

Let $\mathbb{H}^m_{\mathbb{C}}$ and $\mathbb{H}^M_{\mathbb{C}}$ be complex hyperbolic spaces, and let $\Gamma < \mathsf{Isom}(\mathbb{H}^m_{\mathbb{C}})$ be a uniform lattice. Let $\rho: \Gamma \rightarrow \mathsf{Isom}(\mathbb{H}^M_{\mathbb{C}})$ be a convex co-compact representation. Lattice-preservation conjecture. If $2 \leq m \leq M$, then the image of $\rho$ preserves a totally geodesic copy of $\mathbb{H}^m_{\mathbb{C}}$ in $\mathbb{H}^M_{\mathbb{C}}$. This conjecture is known when $M \leq 2m-1$, by work of Cao--Mok and Yue, but remains open in general.