  A necessary condition for the probabilities of a set of events to exhibit
Bell nonlocality or Kochen-Specker contextuality is that the graph of
exclusivity of the events contains induced odd cycles with five or more
vertices, called odd holes, or their complements, called odd antiholes. From
this perspective, events whose graph of exclusivity are odd holes or antiholes
are the building blocks of contextuality. For any odd hole or antihole, any
assignment of probabilities allowed by quantum mechanics can be achieved in
specific contextuality scenarios. However, here we prove that, for any odd
hole, the probabilities that attain the quantum maxima cannot be achieved in
Bell scenarios. We also prove it for the simplest odd antiholes. This leads us
to the conjecture that the quantum maxima for any of the building blocks cannot
be achieved in Bell scenarios. This result sheds light on why the problem of
whether a probability assignment is quantum is decidable, while whether a
probability assignment within a given Bell scenario is quantum is, in general,
undecidable. This also helps to undertand why identifying principles for
quantum correlations is simpler when we start by identifying principles for
quantum sets of probabilities defined with no reference to specific scenarios.
