Parallelogram characterization of large intersections for admissible triples

Let $G$ be a countable discrete abelian group, and let $(r,s,t)$ be an admissible triple of homomorphisms or corresponding group elements. A quadruple $(\varphi_0,\varphi_1,\varphi_2,\varphi_3)$ forms a parallelogram when

$$\varphi_i+\varphi_j=\varphi_k+\varphi_l$$

for some permutation $(i,j,k,l)$ of $(0,1,2,3)$. The triple has the large intersections property in the sense used for triple recurrence.

Parallelogram conjecture. The admissible triple $(r,s,t)$ has the large intersections property if and only if $(0,r,s,t)$ forms a parallelogram.

The paper proves sufficiency for admissible families arising by multiplication by integers and conjectures necessity in general; the condition characterizes quadruples of the form $(0,\varphi,\psi,\varphi+\psi)$ up to reordering and shifts.