Setting and definitions
Let $X = \{X(t), t \in T \}$, $T \subset \mathbb{Z}$, be an infinite-variance associated stochastic process, i.e. $$ \text{Cov}(f(X(I)), g(X(J))) \geq 0 $$ for all finite disjoint subsets $I, J \subset T$ and bounded, coordinate-wise increasing Borel functions $f: \mathbb{R}^{\vert I \vert} \rightarrow \mathbb{R}$, $g: \mathbb{R}^{\vert J \vert} \rightarrow \mathbb{R}$.
A stochastic process $Y$ is called (BL, $\theta$)-dependent if if there exists a non-increasing sequence $\theta = (\theta_r)_{r \in \mathbb{Z}}$ with $\theta_r \rightarrow 0$ as $r \rightarrow \infty$ and $$ \Big\vert \text{Cov} \Big( f\big(Y(I)\big), g\big(Y(J)\big) \Big) \Big\vert \leq \text{Lip}(f)\text{Lip}(g)(\vert I \vert \wedge \vert J \vert) \theta_r $$
for any bounded Lipschitz-continuous functions $f: \mathbb{R}^{\vert I \vert} \rightarrow \mathbb{R}$, $g: \mathbb{R}^{\vert J \vert} \rightarrow \mathbb{R}$ and finite disjoint subsets $I, J \subset T$ such that $\text{dist}(I, J) := \min\{ \vert i - j \vert : i \in I, j \in J \} = r$.
Question
Is $X$ (BL, $\theta$)-dependent? The finite-variance case can be proven as seen below. But the proof relies on covariances of the process and I don't know how to generalize the main inequality that was used in the proof.
Proof for finite-variance case
If $X$ had a finite-variance, then Theorem 5.3. in Bulinski & Shashkin (2007) states that $$ \Big\vert \text{Cov} \Big( f\big(X(I)\big), g\big(X(J)\big) \Big) \Big\vert \leq \sum_{i \in I} \sum_{j \in J} \text{Lip}_i(f)\text{Lip}_j(g) \text{Cov}(X(i), X(j)) $$ for all any bounded Lipschitz-continuous functions $f: \mathbb{R}^{\vert I \vert} \rightarrow \mathbb{R}$, $g: \mathbb{R}^{\vert J \vert} \rightarrow \mathbb{R}$ and finite disjoint subsets $I, J \subset T$. Hence, $X$ is (BL, $\theta$)-dependent with $$ \theta_r := \sup_{i \in I} \sum_{j \in \mathbb{Z} : \vert i - j \vert \geq r} \vert \text{Cov}(X(i), X(j)) \vert $$ under the assumption that these quantities exist and tend to zero.
