Let $\mathcal{R}_d \subset \bar{\mathbb{Q}}$ be the set of all roots of degree $d$ polynomials with $\{-1,0,1\}$ coefficients and $$ c(d) := \min_{\substack{ \alpha, \beta \in \mathcal{R}_d \\ \alpha^k \neq \beta^l}}' \frac{[\mathbb{Q}(\alpha,\beta):\mathbb{Q}]}{[\mathbb{Q}(\alpha):\mathbb{Q}] \cdot [\mathbb{Q}(\beta):\mathbb{Q}]} \leq 1, $$ where the minimum is over all pairs of multiplicatively independent $\alpha, \beta \in \mathcal{R}_d$. This theorem of Amoroso and David implies the lower bound $c(d) \gg (\log{d})^{-20}$. Thus, in particular, it follows easily that there are only finitely many irreducible and non-cyclotomic $\{-1,0,1\}$-polynomials $f(X)$ for which the number field $\mathbb{Q}[X] / (f(X))$ is Galois over $\mathbb{Q}$.

I wondered how small an upper bound on $c(d)$, better than the trivial $c(d) \leq 1$, may one prove or expect to have. In particular: is $c(d)$ actually bounded from below by an absolute positive constant, or goes to $0$ as $d$ increases?