Groups of arbitrary lawlessness growth

For a finitely generated lawless group $\Gamma$ and $n \in \mathbb{N}$, let $\mathcal{A}_{\Gamma} (n)$ be the minimal positive integer $M_n$ such that for all nontrivial reduced words $w$ of length at most $n$ in the free group of fixed rank $k \geq 2$, there exists $\overline{g} \in \Gamma^k$ of word-length at most $M_n$ with $w(\overline{g}) \neq e$. For any unbounded nondecreasing function $f : \mathbb{N} \rightarrow \mathbb{N}$ satisfying some mild assumptions, we construct $\Gamma$ such that the function $\mathcal{A}_{\Gamma}$ is equivalent to $f$. Our result generalizes both a Theorem of the first named author, who constructed groups for which $\mathcal{A}_{\Gamma}$ is unbounded but grows more slowly than any prescribed function $f$, and a result of Petschick, who constructed lawless groups for which $\mathcal{A}_{\Gamma}$ grows faster than any tower of exponential functions.