$T^{-1/4}$-noise for random walks in dynamic environment on $\mathbb Z$

We consider a discrete-time random walk $X_t$ on $\mathbb Z$ with transition probabilities $P(X_{t+1}=x+u\mid X_t=x,\xi)=P_0(u)+c(u;\xi(t,x))$, depending on a random field $\xi =\{\xi(t,x)\colon (t,x)\in\mathbb Z\times\mathbb Z\}$. The variables $\xi(t,x)$ take finitely many values, are i.i.d. and $c(u;\cdot\,)$ has zero average. Previous results show that for small stochastic term the CLT holds almost surely, with dispersion independent of the field. Here we prove that the first correction in the CLT asymptotics is a term of order $T^{-1/4}$ depending on the field, with asymptotically gaussian distribution as $T\to\infty$.