We extend to any maximally entangled state of a bipartite system whose constituents are arbitrarily (but finite) dimensional the result, recently derived for two-dimensional constituents, that hidden variable theories cannot have local parts, i.e., that their local averages cannot differ from the quantum mechanical ones.
@article{arxiv.1203.4687,
title = {A general theorem on local parts of hidden variable models},
author = {GianCarlo Ghirardi and Raffaele Romano},
journal= {arXiv preprint arXiv:1203.4687},
year = {2015}
}