The non-generic Galois group conjecture for random integer matrices

Let $M_n(T)$ be the number of integer $n\times n$ matrices $A$ such that the characteristic polynomial $\chi_A$ of $A$ does not have Galois group the full $S_n$. The notation $M_n(T)$ counts matrices whose entries lie in $[-T,T]$. Non-generic Galois group conjecture. As $T\mathop{\rightarrow}\limits\infty$,

$$M_n(T)\asymp T^{n^2-n+1}\log T.$$

This strengthens Rivin's conjecture that the reducible characteristic polynomials contribute on the order of $T^{n^2-n+1}\log T$, and predicts that the same order governs all matrices whose characteristic polynomial has non-full Galois group. The source gives no resolution of the conjecture.