The fixed-constant-term analogue of van der Waerden's conjecture

Let

$$f(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+b$$

be a polynomial with fixed outer coefficients $a_n=1$ and $a_0=b$, and let $E_{n,b}(H)$ denote the number of such polynomials of height at most $H$ whose Galois group is not the full symmetric group $S_n$. Fixed-constant-term van der Waerden conjecture. For fixed $n\geq 2$ and $b\neq 0$,

$$E_{n,b}(H)\asymp H^{n-2}.$$

This is the analogue of van der Waerden's conjecture for the family of monic degree-$n$ polynomials with fixed nonzero constant term. The paper presents it as a direction for future study; its resolution is not established here.