In this question, a pattern is an finite arrangement of pairwise-disjoint axes-parallel rectangles contained in some larger rectangle. A guillotine pattern (GP) is a pattern in which the rectangles can be separated by a sequence of guillotine cuts, which are axes-parallel cuts going from one edge of an available piece to the opposite edge. The interest in guillotine cuts comes from the process of cutting sheets of material in the industry; see guillotine cutting for more details.
The following arrangement of 5 rectangles is a non-GP, since the rectangles cannot be separated by guillotine cuts - the first cut must cross at least one rectangle:
Obviously, if we add rectangles to a non-GP, it remains non-GP, so it is easy to construct non-GPs of any cardinality.
Define a minimal non-GP as a non-GP of which every subset is a GP. The above pattern is not minimal, since it has a subset of 4 rectangles (the four outer recangles) that is not a GP. This subset, in turn, is a minimal non-GP, since every subset of it contains 3 rectangles that can be separated by guillotine cuts.
QUESTION: What is the largest number of rectangles in a minimal non-GP?