Consider a large, fixed $M>2$. For each $n$, let $\alpha_n$ denote the smallest algebraic integer of degree at most $n$, all of whose Galois conjugates lie in the real interval $(0,M)$.

Is there anything known on the rate at which $\alpha_n$ decays to $0$ as $n\to\infty$ ?

The exponential lower-bound $\alpha_n>\frac1 {M^{n}}$ is not hard to prove, but is it reasonable to expect that $\alpha_n$ will actually decay much more slowly, i.e. like an inverse power of $n$ ?