Polynomial height reduction conjecture for fixed degree and discriminant

Let $f\in\mathbb{Z}[X]$ be a polynomial, respectively a monic polynomial, of degree $n\geq 4$, respectively $n\geq 3$, with discriminant $D\ne 0$. Here $H(g)$ denotes the height of a polynomial $g$, and $GL_2(\mathbb{Z})$-equivalence, respectively $\mathbb{Z}$-equivalence, is the equivalence relation used for general, respectively monic, integral polynomials. Polynomial height reduction conjecture. Then $f$ is $GL_2(\mathbb{Z})$-equivalent, respectively $\mathbb{Z}$-equivalent, to a polynomial, respectively monic polynomial, $g$ such that

$$H(g)\leq c_6(n)|D|^{c_7(n)},$$

where $c_6(n)$ and $c_7(n)$ depend only on $n$. The conjecture predicts polynomial, rather than exponential, dependence on the absolute discriminant; it is motivated by effective bounds available when the splitting field is fixed, while the general case remains open.