  Unextendible product bases (UPBs) provide a versatile tool with various
applications across different areas of quantum information theory. Their
comprehensive characterization is thus of great importance and has been a
subject of vital interest for over two decades now. An open question asks about
the existence of UPBs, which are genuinely unextendible, i.e., they are not
extendible even with biproduct vectors. In other words, the problem is to
verify whether there exist genuinely entangled subspaces (GESs), subspaces
composed solely of genuinely multiparty entangled states, complementary to
UPBs. We solve this problem in the negative for many sizes of UPBs in different
multipartite scenarios. In particular, in the all-important case of equal local
dimensions, we show that there are always forbidden cardinalities for such
UPBs, including the minimal ones corresponding to GESs of the maximal
dimensions.
