The asymptotic dimension of the grand arc graph is infinite

Let $Σ$ be a compact, orientable surface of genus $g$, and let $Γ$ be a relation on $π_0(\partial Σ)$ such that the prescribed arc graph $\mathcal{A}(Σ,Γ)$ is Gromov-hyperbolic and non-trivial. We show that $\operatorname{asdim} \mathcal{A}(Σ,Γ) \geq -χ(Σ) - 1$, from which we prove that the asymptotic dimension of the grand arc graph is infinite. More generally, an arc and curve model on $Σ$ is a graph of simple arc and curves on $Σ$, on which $\operatorname{PMap}(Σ)$ acts by permuting vertices. We prove that any connected, Gromov-hyperbolic cocompact arc and curve model $\mathcal{M}$ has $\operatorname{asdim} \mathcal{M} \geq g - \lceil\frac{1}{2} χ(Σ)\rceil$, and that a broad class of arc and curve models on infinite-type surfaces has infinite asymptotic dimension.