  Correlation self-testing of a theory addresses the question of whether we can
identify the set of correlations realisable in a theory from its performance in
a particular information processing task. Applied to quantum theory it aims to
identify an information processing task whose optimal performance is achieved
only by theories realising the same correlations as quantum theory in any
causal structure. In [Phys. Rev. Lett. 125 060406 (2020)] we introduced a
candidate task for this, the adaptive CHSH game. Here, we analyse the maximum
probability of winning this game in different generalised probabilistic
theories. We show that theories with a joint state space given by the minimal
or the maximal tensor product are inferior to quantum theory, before
considering other tensor products in theories whose elementary systems have
various two-dimensional state spaces. For these, we find no theories that
outperform quantum theory in the adaptive CHSH game and prove that it is
impossible to recover the quantum performance in various cases. This is the
first step towards a general solution that, if successful, will have
wide-ranging consequences, in particular, enabling an experiment that could
rule out all theories in which the set of realisable correlations does not
coincide with the quantum set.
