Hadamard powers and totally positive matrices

Considered are continuous, positive Hadamard powers of entry-wise positive (nonnegative) matrices. Those that are eventually (in the sense of all Hadamard powers beyond some point) totally positive, totally nonnegative, doubly nonnegative and doubly positive are characterized. For example, for matrices with at least four rows and four columns, Hadamard powers greater than one of totally positive matrices need not be totally positive, but they are eventually totally positive.