The Hecke-commensurator conjecture for triangle groups

Let $\Gamma$ be a discrete triangle group of type $(m_1,m_2,\infty)$, and let $G={\rm GL}(2,\mathbb{C})$. Define

$$C(\Gamma):=\left\{g\in G\ \Big|\ |\Gamma\backslash \Gamma g\Gamma|<\infty\right\}.$$

Let $Z(G)\cong \mathbb{C}^*$ be the group of scalar matrices. The Hecke-commensurator conjecture. For a triangle group $\Gamma$ of type $(m_1,m_2,\infty)$, one has

$$C(\Gamma)\not=Z(G)\cdot \Gamma$$

if and only if $\Gamma$ is arithmetic.

This conjecture characterizes arithmetic triangle groups through the size of the associated Hecke correspondence semigroup: non-scalar double-coset elements should occur exactly in the arithmetic case. The source gives no resolution or partial result for the claim.