The asymptotic dimension of the grand arc graph is infinite

Let $\mathrm{\Sigma }$ be a compact, orientable surface of genus $g$, and let $\mathrm{\Gamma }$ be a relation on ${\pi }_0(\mathrm{\partial }\mathrm{\Sigma })$ such that the prescribed arc graph $\mathcal{A}(\mathrm{\Sigma },\mathrm{\Gamma })$ is Gromov-hyperbolic and non-trivial. We show that $\mathrm{asdim}\mathcal{A}(\mathrm{\Sigma },\mathrm{\Gamma })\ge -\chi (\mathrm{\Sigma })-1$, from which we prove that the asymptotic dimension of the grand arc graph is infinite. More generally, an arc and curve model on $\mathrm{\Sigma }$ is a graph of simple arc and curves on $\mathrm{\Sigma }$, on which $\mathrm{PMap}(\mathrm{\Sigma })$ acts by permuting vertices. We prove that any connected, Gromov-hyperbolic cocompact arc and curve model $\mathcal{M}$ has $\mathrm{asdim}\mathcal{M}\ge g-\lceil \frac{1}{2}\chi (\mathrm{\Sigma })\rceil$, and that a broad class of arc and curve models on infinite-type surfaces has infinite asymptotic dimension.