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Unextendible product bases (UPBs) provide a versatile tool with variousapplications across different areas of quantum information theory. Theircomprehensive characterization is thus of great importance and has been asubject of vital interest for over two decades now. An open question asks aboutthe existence of UPBs, which are genuinely unextendible, i.e., they are notextendible even with biproduct vectors. In other words, the problem is toverify whether there exist genuinely entangled subspaces (GESs), subspacescomposed solely of genuinely multiparty entangled states, complementary toUPBs. We solve this problem in the negative for many sizes of UPBs in differentmultipartite scenarios. In particular, in the all-important case of equal localdimensions, we show that there are always forbidden cardinalities for suchUPBs, including the minimal ones corresponding to GESs of the maximaldimensions.

Negative result about the construction of genuinely entangled subspaces from unextendible product bases