  Unextendible product basis is an important object in quantum information
theory and features a broad spectrum of applications, ranging bound entangled
states, quantum nonlocality without entanglement, and Bell inequalities with no
quantum violation. A generalized concept called uncompletable product basis
also attracts much attention. In this paper, we find some unextendible product
bases that are uncompletable product bases in every bipartition, which answers
a 19 year-old open question proposed by DiVincenzo et al. [Commun. Math. Phys.
238, 379 (2003)]. As a consequence, we connect such unextendible product bases
to local hiding of information and give a sufficient condition for the
existence of an unextendible product basis, that is still an unextendible
product basis in every bipartition. Our results advance the understanding of
the geometry of unextendible product bases.
