Sarnak's bounded-clustering conjecture for cofinite Fuchsian groups

Let $\Gamma$ be a cofinite Fuchsian group. Its trace set, denoted $\mathrm{Tr}(\Gamma)$, is the set of traces of elements of $\Gamma$, defined up to sign. A set of complex numbers has the bounded clustering property (or B-C property) if there is a constant $K_A$ such that, for every $m,n\in\mathbb Z$, its intersection with each unit square $S(m,n)$ contains fewer than $K_A$ elements, where

$$S(m,n)=\left\{z\in\mathbb C\mid m\leq\operatorname{Re}(z)\leq m+1,\ n\leq\operatorname{Im}(z)\leq n+1\right\}.$$

Also define

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

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

Luo and Sarnak had shown that the trace set of an arithmetic Fuchsian group satisfies the B-C property, and the conjecture proposes converses of this phenomenon. The source does not state whether these assertions have been resolved.