Binomial-coefficient random polynomial reducibility conjecture

Let $h_{\mathrm{Bin},K,d}(x)=x^d+a_{d-1}x^{d-1}+\dots+a_1x+a_0$ be a random monic polynomial whose coefficients $a_0,\dots,a_{d-1}$ are independently chosen from $[0,K]$ according to the $\operatorname{Bin}(K,1/K)$ distribution. Binomial reducibility conjecture. The probability that $h_{\mathrm{Bin},K,d}(x)$ is reducible, divided by the probability that its constant coefficient is zero, goes to $1$ as $K$ goes to infinity. This is proposed as a case of the paper's universality heuristic, in a model where the zero constant coefficient supplies the dominant obvious source of reducibility.