Schmutz's linear-growth conjecture for trace sets of cofinite Fuchsian groups

Let $\Gamma$ be a cofinite Fuchsian group. Its trace set is denoted by $\mathrm{Tr}(\Gamma)$. A subset $A$ of the reals has linear growth if there exist positive real constants $C$ and $D$ such that, for every $n$,

$$\#\{a\in A:|a|\leq n\}\leq Cn+D.$$

Schmutz's conjecture. If $\mathrm{Tr}(\Gamma)$ has linear growth, then $\Gamma$ is arithmetic.

This strengthens the bounded-clustering criterion by replacing it with a global linear-growth condition. The source explicitly states that the conjecture remains open, even for non-uniform lattices.