On the extension of the FKG inequality to n functions

The 1971 Fortuin-Kasteleyn-Ginibre inequality for two monotone functions on a distributive lattice is well known and has seen many applications in statistical mechanics and other fields of mathematics. In 2008, one of us (Sahi) conjectured an extended version of this inequality for all n > 2 monotone functions on a distributive lattice. Here, we prove the conjecture for two special cases: for monotone functions on the unit square in Rk whose upper level sets are k-dimensional rectangles and, more significantly, for arbitrary monotone functions on the unit square in R2. The general case for Rk,k>2, remains open.