  The study of nonlocality in scenarios that depart from the bipartite
Einstein-Podolsky-Rosen setup is allowing to uncover many fundamental features
of quantum mechanics. Recently, an approach to building network-local models
based on machine learning lead to the conjecture that the family of quantum
triangle distributions of [arXiv:1905.04902] did not admit triangle-local
models in a larger range than the original proof. We prove part of this
conjecture in the affirmative. Our approach consists in reducing the family of
original, four-outcome distributions to families of binary-outcome ones, and
then using the inflation technique to prove that these families of
binary-outcome distributions do not admit triangle-local models. This
constitutes the first successful use of inflation in a proof of quantum
nonlocality in networks whose nonlocality could not be proved with alternative
methods. Moreover, we provide a method to extend proofs of network nonlocality
in concrete distributions of a parametrized family to continuous ranges of the
parameter. In the process, we produce a large collection of network Bell
inequalities for the triangle scenario with binary outcomes, which are of
independent interest.
