Asymptotic count of reducible integer matrices

Let $N(n,B)$ denote the number of $n\times n$ integer matrices whose entries have absolute value at most $B$ and whose characteristic polynomial is reducible over $\mathbb{Z}$. For some nonzero constants $c_1,c_2$, the source establishes bounds of the form

$$c_1 B^{n^2-n+1}\log B\leq N(n,B)\leq c_2 B^{n^2-1}\log B.$$

Asymptotic counting conjecture. The lower bound gives the true order of growth:

$$N(n,B)\asymp c_n B^{n^2-n+1}\log B.$$

For $n=2$, the upper and lower bounds already have the same order, whereas for larger $n$ the conjecture would close the gap between the available bounds.