Before revision:  The problem of assessing the sign-stability of the convex full of a finite  family of Metzler matrices is also solved, and a necessary and sufficient condition for the existence of a common Lyapunov function for all the matrices in the convex hull is obtained.
After revision:  The problem of assessing the sign-stability of the convex full of a finite and summable family of Metzler matrices is also solved, and a necessary and sufficient condition for the existence of a common Lyapunov function for all the matrices in the convex hull is obtained.