  The determination of genuine entanglement is a central problem in quantum
information processing. We investigate the tripartite state as the tensor
product of two bipartite entangled states by merging two systems. We show that
the tripartite state is a genuinely entangled state when the range of both
bipartite states are entanglement-breaking subspaces. We further investigate
the tripartite state when one of the two bipartite states has rank two. Our
results provide the latest progress on a conjecture proposed in the paper [Yi
Shen $\textit{et al}$, J. Phys. A 53, 125302 (2020)]. We apply our results to
construct multipartite states whose bipartite reduced density operators have
additive EOF. Further, such states are distillable across every bipartition
under local operations and classical communications.
