Galois groups of random polynomials of large degree

We study random polynomials of the form $R(x)=x^n+ω_{n-1}x^{n-1}+\cdots+ω_0$, where $ω_0,\dots,ω_{n-1}$ are independent, uniformly bounded integer-valued random variables, and $ω_1,\dots,ω_{n-1}$ have a fixed common law $μ$. We prove (unconditionally) that, if the Rényi entropy of order $2$ satisf