Eventual non-negativity of Matrices

Consider the problem: given a matrix $M$ over integers, does there exist a natural number $n$ such that $M^n$ has only non-negative entries? Is this question as hard as ultimate positivity? Note that if you consider two matrices $M, N$ and ask if there exists $n$ such that $M^n+ N^n$ has only non-negative entries, that problem is indeed equivalent to ultimate positivity, but with a single matrix, we do not know. Note also that if we ask strict positivity of entries, the single matrix case becomes easy!