Not all GKK $\tau$-matrices are stable

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, © 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 ©. It was conjectured that: (A), (B) implies (D) [D. Carlson, J. Res. Nat. Bur. Standards Sect. B 78 (1974) 1-2], (A), © 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), © 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.