Exponential memory decay for deterministic walks in random environment

Let $z=(z(n))_{n\geq 0}$ be a path in the admissible path space, let $z^k=z(1),\ldots,z(k-1)$ denote its past, and let $\widehat z_m(k)=z(m+k)-z(m)$ with $\widehat z^k=\widehat z_m(1),\ldots,\widehat z_m(k-1)$. Let $\tau_{z(m)}$ denote the environment shift by $z(m)$. Exponential memory-decay conjecture. Under some appropriate technical conditions on $\mathbb P$ and $h_0$, there exist $\nu\in(0,1)$ and C_\\#>0 such that, for all $\bar\omega\in\Omega$,

\left|\mathbb P_\star(z(n)\mid z^n,\bar\omega)-\mathbb P_\star(\widehat z_m(n-m)\mid\widehat z^{n-m},\tau_{z(m)}\bar\omega)\right|\leq C_\\#\nu^{n-m}.

This asserts that the conditional jump probabilities have exponentially weak dependence on the remote past, providing the expected Gibbsian or weak-memory structure for the non-Markovian process generated by the deterministic walk.