A local barycentric version of the Bak-Sneppen model

We study the behaviour of the interacting particle system, arising from the Bak-Sneppen model and Jante's law process. Let $N$ vertices be placed on a circle, such that each vertex has exactly two neighbours. To each vertex assign a real number, called {\em fitness}. Now find the vertex which fitness deviates most from the average of the fitnesses of its two immediate neighbours (in case of a tie, draw uniformly among such vertices), and replace it by a random value drawn independently according to some distribution $\zeta$. We show that in case where $\zeta$ is a uniform or a discrete uniform distribution, all the fitnesses except one converge to the same value.