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!