Common and Sidorenko equations in Abelian groups

A linear configuration is said to be common in a finite Abelian group G if for every 2-coloring of G the number of monochromatic instances of the configuration is at least as large as for a randomly chosen coloring. Saad and Wolf conjectured that if a configuration is defined as the solution set of a single homogeneous equation in an even number of variables over G, then it is common in F-p(n) if and only if the equation's coefficients can be partitioned into pairs that sum to zero mod p. This was proven by Fox, Pham and Zhao for sufficiently large n. We generalize their result to all sufficiently large Abelian groups G for which the equation's coefficients are coprime to |G|.