Conjectured asymptotic bounds for the probability of alternating Galois group

Fix $n\geq4$. Let $f$ be a degree-$n$ polynomial whose coefficients are chosen independently and uniformly from $[-L,L]\cap\mathbb{Z}$, and let $G_f$ denote its Galois group. The alternating-Galois-group conjecture. As $L\to\infty$,

$$L^{-n/2+\epsilon}\gg\operatorname{Prob}(G_f=A_n)\gg L^{-n/2}.$$

Here the implied constants may depend on $n$ and $\epsilon$, as is standard for the notation $\gg$. The conjecture refines the known upper bound discussed in the source and is motivated by proved lower bounds and the small-degree cases, but the supplied text gives no resolution status for the general assertion.