Nonexistence of $n$-qubit unextendible product bases of size $2^n-5$

It is known that the $n$-qubit system has no unextendible product bases (UPBs) of cardinality $2^n-1$, $2^n-2$ and $2^n-3$. On the other hand the $n$-qubit UPBs of cardinality $2^n-4$ exist for all $n\ge 3$. We prove that they do not exist for cardinality $2^n-5$.