Quasi-isometric embeddings inapproximable by Anosov representations

We construct examples of quasi-isometric embeddings of word hyperbolic groups into $\mathsf{S}\mathsf{L}(d,\mathbb{R})$ for $d⩾5$ which are not limits of Anosov representations into $\mathsf{S}\mathsf{L}(d,\mathbb{R})$. As a consequence, we conclude that an analogue of the density theorem for $\mathsf{P}\mathsf{S}\mathsf{L}(2,\mathbb{C})$ does not hold for $\mathsf{S}\mathsf{L}(d,\mathbb{R})$ when $d⩾5$.