Conjecture on the number of generalized degenerate monic integer polynomials

Let $M_n(H)$ be the set of generalized degenerate monic integer polynomials of degree $n$ and height at most $H$. Growth conjecture. For every integer $n\ge 2$,

$$\# M_n(H) \asymp H^{n-1}.$$

The preceding results establish this growth rate for some degrees, including $n=4$ and prime degrees, while the conjecture asserts that the logarithmic factor in the general upper bound can be removed for every $n\ge 2$.