Admissibility characterization for pair large intersections

Let $G$ be a countable discrete abelian group, and let $\varphi,\psi:G\to G$ be homomorphisms such that

$$\{g\in G:\varphi(g)=\psi(g)\}$$

has infinite index in $G$. A pair is admissible when it satisfies the paper's admissibility condition, and it has the large intersections property when its associated triple intersections are uniformly large on a syndetic subset of $G$.

Pair characterization conjecture. The pair $\{\varphi,\psi\}$ has the large intersections property if and only if it is admissible.

This conjecture gives evidence that admissibility is essential: the paper explains that non-admissible families provide failures of large intersections, whereas the admissible case is supported by the characteristic-factor and ergodic arguments developed earlier.