Van der Waerden's conjecture on exceptional Galois groups of polynomials

Let $f(X)=X^n+a_1X^{n-1}+3+a_{n-1}X+a_n$ be a monic polynomial of degree $n\geqslant 3$ with integer coefficients, and let $G_f$ be its Galois group, viewed as a subgroup of the symmetric group $S_n$. For a large positive real number $H$, define

$$E_n(H):=\#\{(a_1,\ldots,a_n)\in(\mathbb Z\cap[-H,H])^n:G_f\not\simeq S_n\}$$

and let $R_n(H)$ be the number of monic reducible polynomials of degree $n$ with integer coefficients in $[-H,H]$.

Van der Waerden's conjecture. For $n\geqslant 3$, we have

$$E_n(H)=R_n(H)(1+o(1)).$$

Equivalently, monic irreducible polynomials of degree $n$ whose Galois group is not $S_n$ should number $o(H^{n-1})$ as $H\to\infty$. The conjecture predicts that the exceptional polynomials are asymptotically accounted for by reducible polynomials; the paper establishes the first two solved cases, for cubics and quartics, while the general statement remains open.