The zero-density conjecture for square systems over free abelian groups

Let $F_{1,n,n}(r)$ denote the sum of the reciprocals of the absolute determinants of all full-rank $n\times n$ integer matrices whose entries have Euclidean norm at most $r$, and let $SAT(\mathbb{Z}^m,n,n)$ be the set of satisfiable systems of $n$ equations in $n$ variables over $\mathbb{Z}^m$. Zero-density conjecture. One has

$$F_{1,n,n}(r) = O(r^{n^2-n}\ln(r))$$

and

$$\rho(SAT(\mathbb{Z}^m,n,n))=0.$$

The preceding discussion motivates the conjecture by combining fixed-determinant matrix-counting asymptotics with the relationship between the function $F_{m,k,n}$ and the asymptotic density of satisfiable systems. The stated determinant estimate is presented heuristically under the assumption that determinant values occur roughly equally often, so the conjecture remains open in the supplied text.