  The boundary between classical and quantum correlations is well characterised
by linear constraints called Bell inequalities. It is much harder to
characterise the boundary of the quantum set itself in the space of
no-signaling correlations. For the points on the quantum boundary that violate
maximally some Bell inequalities, Oppenheim and Wehner [Science 330, 1072
(2010)] pointed out a complex property: the optimal measurements of Alice steer
Bob's local state to the eigenstate of an effective operator corresponding to
its maximal eigenvalue. This effective operator is the linear combination of
Bob's local operators induced by the coefficients of the Bell inequality, and
it can be interpreted as defining a fine-grained uncertainty relation. It is
natural to ask whether the same property holds for other points on the quantum
boundary, using the Bell expression that defines the tangent hyperplane at each
point. We prove that this is indeed the case for a large set of points,
including some that were believed to provide counterexamples. The price to pay
is to acknowledge that the Oppenheim-Wehner criterion does not respect
equivalence under the no-signaling constraint: for each point, one has to look
for specific forms of writing the Bell expressions.
