Non-ergodic counterexample to large intersections for admissible pairs

Let $G$ be a countable discrete abelian group, and let $\varphi,\psi:G\to G$ be an admissible pair of homomorphisms. A measure-preserving system is a quadruple $(X,\mathcal B,\mu,(T_g)_{g\in G})$, and $A\in\mathcal B$ is a measurable set. The pair has the large intersections property when the associated triple intersections satisfy the relevant uniform lower bound on a syndetic subset of $G$.

Non-ergodic failure conjecture. There exist a necessarily non-ergodic measure-preserving system $(X,\mathcal B,\mu,(T_g)_{g\in G})$, a set $A\in\mathcal B$ with $\mu(A)>0$, and $c<1$ such that

$$\mu\left(A\cap T_{\varphi(g)}^{-1}A\cap T_{\psi(g)}^{-1}A\right)\le c\mu(A)^3$$

for every $g\ne0$.

The conjecture says that ergodicity, or a finite ergodic decomposition, is essential for the large-intersection conclusions established in the paper.