Before revision:  It is first established that checking the sign-stability of a Metzler sign-matrix can be either characterized in terms of the Hurwitz stability of the unit sign-matrix in the corresponding qualitative class, or in terms of the acyclicity of the graph associated with the sign-pattern . Similar results are obtained for the case of  block-matrices and  mixed-matrices, the latter containing both sign patterns and fixed real entries.
After revision:  It is first established that checking the sign-stability of a Metzler sign-matrix can be either characterized in terms of the Hurwitz stability of the unit sign-matrix in the corresponding qualitative class, or in terms the negativity of the diagonal elements of the Metzler sign-matrix and the acyclicity of the graph associated with the sign-pattern . Similar results are obtained for the case of block-matrices and mixed-matrices, the latter containing both sign patterns and fixed real entries.