The asymptotic dimension of the grand arc graph is infinite

Let $\Sigma$ be a compact, orientable surface of genus $g$, and let $\Gamma$ be a relation on $\pi_0(\partial \Sigma)$ such that the prescribed arc graph $\mathcal{A}(\Sigma,\Gamma)$ is Gromov-hyperbolic and non-trivial. We show that $\operatorname{asdim} \mathcal{A}(\Sigma,\Gamma) \geq -\chi(\Sigma) - 1$, from which we prove that the asymptotic dimension of the grand arc graph is infinite. More generally, an arc and curve model on $\Sigma$ is a graph of simple arc and curves on $\Sigma$, on which $\operatorname{PMap}(\Sigma)$ acts by permuting vertices. We prove that any connected, Gromov-hyperbolic cocompact arc and curve model $\mathcal{M}$ has $\operatorname{asdim} \mathcal{M} \geq g - \lceil\frac{1}{2} \chi(\Sigma)\rceil$, and that a broad class of arc and curve models on infinite-type surfaces has infinite asymptotic dimension.