Uniform upper-bound conjecture for matrices with a given characteristic polynomial

Let $R_n(H;f)$ denote the number of matrices $A\in\mathcal M_n(\mathbb Z;H)$ with characteristic polynomial $f\in\mathbb Z[X]$, where $\mathcal M_n(\mathbb Z;H)$ is the set of $n\times n$ integer matrices whose entries have absolute value at most $H$. Uniform upper-bound conjecture. Uniformly over polynomials $f$,

$$R_n(H;f)\le H^{n(n-1)/2+o(1)}$$

as $H\to\infty$. The conjecture seeks a bound uniform in the coefficients of $f$ and applies without requiring $f$ to be irreducible. Existing asymptotic results for irreducible or otherwise restricted characteristic polynomials motivate it, but do not provide the required uniform estimate for arbitrary $f$.