Sarnak's trace-set arithmeticity conjecture for cofinite Fuchsian groups

Let $\Gamma$ be a cofinite Fuchsian group. Its trace set $\mathrm{Tr}(\Gamma)$ is the set of traces of elements of $\Gamma$, defined up to sign. A set $A$ of real numbers has the bounded clustering property if there is a constant $K_A$ such that $A\cap[n,n+1]$ has fewer than $K_A$ elements for every $n\in\mathbb Z$, and define

$$\operatorname{Gap}(A):=\inf\{|a-b|:a,b\in A,\ a\ne b\}.$$

Sarnak's conjecture. If $\mathrm{Tr}(\Gamma)$ satisfies the bounded clustering property, then $\Gamma$ is arithmetic. If $\operatorname{Gap}(\mathrm{Tr}(\Gamma))>0$, then $\Gamma$ is derived from a quaternion algebra.

Luo and Sarnak proved the bounded clustering property for arithmetic Fuchsian groups; the conjectured converses relate trace-set sparsity to arithmeticity and quaternionic origin. The resolution status of these two assertions is not specified in the source.