Surfaces without quasi-isometric simplicial triangulations

We construct a complete Riemannian surface $\Sigma$ that admits no triangulation $G\subset \Sigma$ such that the inclusion $G^{(1)} \hookrightarrow \Sigma$ is a quasi-isometry, where $G^{(1)}$ is the simplicial 1-skeleton of $G$. Our construction is without boundary, has arbitrarily large systole, and furthermore, there is no embedded graph $G\subset\Sigma$ such that $G^{(1)} \hookrightarrow \Sigma$ is a quasi-isometry. This answers a question of Georgakopoulos.