  Multipartite entanglement is an indispensable resource in quantum
communication and computation, however, it is a challenging task to faithfully
quantify this global property of multipartite quantum systems. In this work, we
study the concurrence fill, which admits a geometric interpretation to measure
genuine tripartite entanglement for the three-qubit system in [S. Xie {\it et
al.}, Phys. Rev. Lett. \textbf{127}. 040403 (2021)]. First, we use the
well-known three-tangle and bipartite concurrence to reformulate this
quantifier for all pure states. We then construct an explicit example to
conclusively show the concurrence fill can be increased under local operation
and classical communications (LOCCs) {\it on average}, implying it is not an
entanglement monotone. Moreover, we give a simple proof of the
LOCC-monotonicity of three-tangle and find that the bipartite concurrence and
the squared can have distinct performances under the same LOCCs. Finally, we
propose a reliable monotone to quantify genuine tripartite entanglement, which
can also be easily generalised to the multipartite system. Our results shed
light on studying genuine entanglement and also reveal the complex structure of
multipartite systems.
