  By considering the nonuniform exponential dichotomy spectrum, we introduce a
global asymptotic nonuniform stability conjecture for nonautonomous
differential systems, whose restriction to the autonomous case is related to
the classical Markus--Yamabe Conjecture: we prove that the conjecture is
verified for a family of triangular systems of nonautonomous differential
equations satisfying boundedness assumptions. An essential tool to carry out
the proof is a necessary and sufficient condition ensuring the property of
nonuniform exponential dichotomy for upper block triangular linear differential
systems. We also obtain some byproducts having interest on itself, such as, the
diagonal significance property in terms on the above mentioned spectrum.
