  It remains an open question whether every pure multipartite state that is
genuinely entangled is also genuinely nonlocal. Recently, a new general
construction of Bell inequalities allowing the detection of genuine
multipartite nonlocality (GMNL) in quantum states was proposed in [F. J.
Curchod, M. L. Almeida, and A. Acin, New J. Phys. 21, 023016 (2019) with the
aim of addressing the above problem. Here we show how, in a simple manner, one
can improve this construction to deliver tighter Bell inequalities for
detection of GMNL. Remarkably, we then prove one of the improved Bell
inequalities to be powerful enough to detect GMNL in every three-qubit
genuinely entangled state. We also generalize some of these inequalities to
detect not only GMNL but also nonlocality depth in multipartite states and we
present a possible way of generalizing them to the case of more outcomes.
