  It was recently found that the indefinite causal order in the quantum switch
can be certified device-independently when assuming the impossibility of
superluminal influences. Here we strengthen this result in two ways. First, we
give a proof of this fact which is possibilistic, rather than probabilistic,
i.e. which does not rely on the validity of probability theory at the hidden
variable level. Then, returning to the probabilistic setting, we show that the
indefinite causal order in the quantum switch is also maximal, in the sense
that the observed correlations are incompatible even with the existence of a
causal order on only a small fraction of the runs of the experiment. While the
original result makes use of quantum theory's violation of a
Clauser-Horne-Shimony-Holt inequality, the proofs presented here are based on
Greenberger, Horne, and Zeilinger's and Mermin's proofs of nonlocality,
respectively.
