Guralnick's conjecture on monodromy groups of large-degree indecomposable coverings

Let $f:X\rightarrow \mathbb P^1$ be an indecomposable covering of degree $n$, with $X$ of genus $g_X=g$, and let $\operatorname{Mon}(f)$ denote its monodromy group. For each integer $g\geq 0$, there exists a constant $N_g$ such that, whenever $n\geq N_g$, one has one of the following possibilities:

$$\operatorname{Mon}(f)=A_n\text{ or }S_n;$$ $$\operatorname{Mon}(f)=A_d\text{ or }S_d\quad\text{with }n=d(d-1)/2\text{ and }g=0;$$ $$A_d^2\leq \operatorname{Mon}(f)\leq (S_d^2)\rtimes C_2,$$

where $n=d^2$, the semidirect action of $C_2$ permutes the two copies of $S_d$, and $g\leq 1$; or

$$\operatorname{Mon}(f)\leq (C_p)^i\rtimes C_k,$$

with $i\in\{1,2\}$, $k\in\{1,2,3,4,6\}$, and $p$ prime, where $n=p^i$ and $g\leq 1$.