Violation of the Finner inequality in the four-output triangle network

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 four outputs per party is especially intriguing. The ``elegant distribution'' [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-signaling distributions with independent sources (NSI distributions). Here we provide evidence that this conjecture is false by constructing a four-output network box that violates 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.