Before revision:  In this we are inspired by the analogous results for ]  P-matrices (matrices with all principal minors positive) .
After revision:  We do away with the former, and show that A is TP_k if and only if every contiguous submatrix of size \leq k has the sign non-reversal property. In fact this can be strengthened to only consider test vectors in \mathbb{R ] P-matrices (matrices with all principal minors positive) .