Hitchin's conjecture on stabilizers of invariant quadruples of torsion points

Let $S=\{(r_i,\theta_i)\}_{i=1}^{4}\in\mathcal{R}_4$ have common order $n$. Let $\Gamma_S=\{\gamma^{-T}\in\operatorname{SL}_2(\mathbb{Z}):\gamma\cdot S=S\}$ be the stabilizer of $S$, and let $\Delta_S=\{\gamma\in\operatorname{SL}_2(\mathbb{Z}):\mu_S(\gamma\cdot\tau)=\mu_S(\tau)\text{ for any }\tau\in X(n)\}$ be the stabilizer of the associated meromorphic invariant $\mu_S$. The groups are considered up to the $\operatorname{SL}_2(\mathbb{Z})$-equivalence in $\mathcal{R}_4$. Hitchin's conjecture. For all but finitely many $S$, one has

$$\Gamma_S=\Delta_S.$$

Theorem 1 classifies the quadruples for which $\mu_S$ is constant, resolving Hitchin's preceding conjecture; the finiteness assertion for equality of the two stabilizers is the next natural question and is not resolved here.