  Network nonlocality allows one to demonstrate nonclassicality in networks
with fixed joint measurements, that is without random measurement settings. The
simplest network in a loop, the triangle, with 4 outputs per party is
especially intriguing. The "elegant distribution" [N. Gisin, Entropy 21, 325
(2019)] still resists analytic proofs, despite its many symmetries. In
particular, this distribution is invariant under any output permutation. The
Finner inequality, which holds for all local and quantum distributions, has
been conjectured to be also valid for all no-signalling distributions with
independent sources (NSI distributions). Here we provide evidence that this
conjecture is false by constructing a 4-output network box that violate the
Finner inequality and prove that it satisfies all NSI inflations up to the
enneagon. As a first step toward the proof of the nonlocality of the elegant
distribution, we prove the nonlocality of the distributions that saturates the
Finner inequality by using geometrical arguments.
