Cyclic Galois group conjecture for generic spectral covers

Let $X$ be a compact Riemann surface of genus $g_X$, and let $L\to X$ be a holomorphic line bundle of positive degree. Fix a generic section

$$s\in H^0(X,L^r)$$

with distinct zeros, and consider the smooth integral cyclic spectral cover $\pi:X_s\to X$ and the induced extension of function fields. Cyclic Galois group conjecture. This extension is Galois, and the Galois group of the cover $\pi$ is cyclic of order $r$. The conjecture concerns the persistence of the expected cyclic Galois structure for generic spectral covers over compact Riemann surfaces; the source notes that it is believed to hold in higher genus but gives no definite reference for a proof.