Amoroso's height conjecture for generators of symmetric Galois extensions

Let $\alpha\in\overline{\mathbb{Q}}$ generate a Galois extension of degree $d=n!$ over $\mathbb{Q}$ whose Galois group is isomorphic to the symmetric group $\mathfrak{S}_n$. Here $h(\alpha)$ denotes the absolute logarithmic Weil height. Amoroso's conjecture. There is a function $c(d)$ tending to infinity with $d$ such that

$$h(\alpha)\geq c(d).$$

This conjecture seeks a height lower bound for generators of symmetric Galois extensions that grows with the extension degree. The supplied text presents it as a conjecture motivated by known results for particular generators, and gives no resolution.