The martingale existence conjecture for attraction-repulsion processes

Let $f$ be an interaction function satisfying the conditions of Theorem 2, and let $n$ be finite. A function

$$h:[0,1]^n\rightarrow\mathbb{R}$$

should exist such that $\{h(\Phi_t)\}_{t\in\mathbb{N}}$ is a non-negative super-martingale or a bounded sub-martingale with respect to the canonical filtration, and such that for every $\varepsilon>0$ the condition

$$\lvert h(\Phi_{t+1})-h(\Phi_t)\rvert<\varepsilon$$

implies that the process is within $\delta(\varepsilon)$ of a trivial state, where $\delta(\varepsilon)\to0$ as $\varepsilon\to0^+$. This would establish the martingale criterion for almost-sure trivialization.