Van der Waerden proved for monic irreducible polynomials $f$ of degree $n$ with bounded height $X$:

$$
|\{ f(x)\in \mathbb{Z}[x]: Gal(f)\neq S_n \}|=o(X)
$$
Where height of $f$ is defined by maximum of absolute value of its coefficient.

So if we order monic irreducible polynomial of degree $n$ with respect to height, then we expect to obtain symmetric group as its Galois group. Now what if we want to replace height by discriminant.

Do we have any table or any conjecture about this? More precisely is it true:

$$ |\{ f\in \mathbb{Z}[x]: monic, irreducible \deg(f)=n, |Disc(f)|\leq X, Gal(f)\neq S_n\}|=o(X) $$