Failure of large intersections for quadruple recurrence

Let $G$ be a countable discrete abelian group, and let $\varphi_1,\dots,\varphi_k:G\to G$ be distinct, nonzero homomorphisms. Assume, as in the source's nontriviality qualification, that

$$\{g\in G:\varphi_i(g)=\varphi_j(g)\}$$

has infinite index in $G$ for every $i\ne j$. A family has the large intersections property when its associated multiple intersections are uniformly large on a syndetic subset of $G$.

Quadruple-recurrence conjecture. If $k\ge4$, then $\{\varphi_1,\dots,\varphi_k\}$ does not have the large intersections property.

The paper suspects that Khintchine-type large-intersection results fail for patterns of length five and longer, with the stated threshold $k\ge4$ covering the corresponding quadruple-recurrence formulation.