  Let $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$. For all but finitely many $m\in
\mathbb{N}$, we exhibit the first examples of non-locally rigid, Zariski dense,
robust quasi-isometric embeddings of hyperbolic groups in
$\mathsf{SL}_m(\mathbb{K})$ which are not limits of Anosov representations. As
a consequence, we show that an analogue of Sullivan's structural stabilty
theorem for Kleinian groups fails for Anosov representations in
$\mathsf{SL}_m(\mathbb{C}), m\geq 30$.
