  Hermitian positive definite, totally positive, and nonsingular M-matrices
enjoy many common properties, in particular:
  (A) positivity of all principal minors,
  (B) weak sign symmetry,
  (C) eigenvalue monotonicity,
  (D) positive stability.
  The class of GKK matrices is defined by properties (A) and (B), whereas the
class of nonsingular $\tau$-matrices by (A) and (C). It was conjectured that:
  (A), (B) implies (D) [D. Carlson, J. Res. Nat. Bur. Standards Sect. B 78
(1974) 1-2],
  (A), (C) implies (D) [G.M. Engel and H. Schneider, Linear and Multilinear
Algebra 4 (1976) 155-176],
  (A), (B) implies a property stronger than (D) [R. Varga, Numerical Methods in
Linear Algebra, 1978, pp. 5-15],
  (A), (B), (C) implies (D) [D. Hershkowitz, Linear Algebra Appl. 171 (1992)
161-186].
  We describe a class of unstable GKK $\tau$-matrices, thus disproving all four
conjectures.
