The monodromy group conjecture for toric surfaces

Let $X$ be a smooth toric surface and let $\mathcal L$ be an ample line bundle on $X$ whose generic fiber is not hyperelliptic. Assume $r>1$ or $g(\mathcal L)\geq 5$, and let $\Gamma_{\mathcal L}$ be the monodromy group of the associated family of curves, acting on a surface $\Sigma_{g(\mathcal L)}$. Let $\phi_{\mathcal L}$ be the associated $\mathbb Z/(r-1)\mathbb Z$-valued spin structure, and write $\operatorname{Mod}(\Sigma_{g(\mathcal L)})[\phi_{\mathcal L}]$ for its stabilizer. The monodromy group conjecture. For any pair $(X,\mathcal L)$ as above, there is an equality

$$\Gamma_{\mathcal L}=\operatorname{Mod}(\Sigma_{g(\mathcal L)})[\phi_{\mathcal L}].$$

The theorem stated in the source proves a more nuanced result: equality holds when $r$ is odd, whereas for even $r$ the monodromy group is only asserted to be a finite-index subgroup containing the admissible twist subgroup. The conjecture was formulated independently by the author and by Cr\etois--Lang; the source presents its toric-surface theorem as addressing it.