The deformation conjecture for cocompact hyperbolic lattices

For $n\geqslant 2$, let $\Gamma$ be a cocompact lattice of $\operatorname{O}(n,1)$, and let $N\geqslant n$. Consider the standard embedding

$$\Gamma\hookrightarrow\operatorname{O}(n,1)\hookrightarrow\operatorname{O}(N,1).$$

A representation is strictly dominated by this embedding if it is dominated in the sense used in the source. Deformation conjecture. For every $n\geqslant 2$, there exist a cocompact lattice $\Gamma$ of $\operatorname{O}(n,1)$ and $N\geqslant n$ such that every neighbourhood of the standard embedding in $\operatorname{Hom}(\Gamma,\operatorname{O}(N,1))$ contains a representation strictly dominated by it. The source presents this as a sufficient conjectural statement that would imply the preceding cocompact hyperbolic-lattice conjecture.