Finitary Khintchine-type recurrence conjecture for finite abelian groups

Let $G$ be a finite abelian group of order $N$, let $\textup{End}(G)$ denote its endomorphism ring, and let $A\subseteq G$. For $\alpha,\varepsilon>0$, write $N_0=N_0(\alpha,\varepsilon)$ for a threshold depending only on these parameters. Finitary recurrence conjecture. For every $\alpha,\varepsilon>0$, there exists $N_0=N_0(\alpha,\varepsilon)$ such that, whenever $N\geq N_0$, $\varphi,\psi\in\textup{End}(G)$ satisfy that $\psi-\varphi$ is an automorphism, and $|A|\geq\alpha N$, there exists $y\in G\setminus\{0\}$ such that

$$\left|\left\{x\in G:\{x,x+\varphi(y),x+\psi(y)\}\subseteq A\right\}\right|>(\alpha^3-\varepsilon)N.$$

This is the proposed finitary analogue of the infinite-group recurrence result, which gives syndetically many parameters with almost the expected density of three-term configurations. Its status is not resolved in the supplied text.