Standard Simplex Conjecture

Prove that for 0 ≤ ρ ≤ 1, among all partitions of R^n into q parts (3 ≤ q ≤ n+1) each of equal Gaussian measure, the maximal noise stability at correlation ρ is attained by a standard simplex partition; furthermore, for −1 ≤ ρ ≤ 0, the standard simplex partition minimizes noise stability.

Background

The conjecture posits that simplex-partition structures optimize Gaussian noise stability for multi-partitions, generalizing the two-part (halfspace) case. It has deep implications in hardness of approximation and social choice, including the Plurality Is Stablest program.

Known implications link it to the Plurality Is Stablest Conjecture, which in turn implies the Standard Simplex Conjecture in a subrange of ρ, but the full conjecture remains unsettled.