[$\bf 2 $]{} (8.5,-0.05) node\[anchor=north\] [$\bf n\!-\!1$]{}; (0.5,0.8) node\[anchor=south\][$\rho (1+\frac{E}{n^\gamma})$]{}; (0.5,-0.8) node\[anchor=north\][$1-\rho$]{}; (4.5,0.8) node\[anchor=south\][$1+\frac{E}{n^\gamma}$]{}; (8.5,-0.8) node\[anchor=north\][$\rho$]{}; (8.5,0.8) node\[anchor=south\][$(1-\rho) (1+\frac{E}{n^\gamma})$]{}; (4.5,-0.8) node\[anchor=north\][$1$]{};

Throughout the paper, a time horizon line $T>0$ is fixed. We are interested in the evolution of this exclusion process in the diffusive time scale $tn^2$, thus we denote by $\{\eta^n_{tn^2}\,; \, t\in [0,T]\}$ the Markov process on $\Omega_n$ associated to the accelerated generator $n^2 \mathcal{L}_n$. The path space of càdlàg trajectories with values in $\Omega_n$ is denoted by $\mathcal{D}([0,T],\Omega_n)$. For any initial probability measure $\mu$ on $\Omega_n$, we denote by ${{\mathbb P}}_{\mu}$ the probability measure on $\mathcal{D}([0,T],\Omega_n)$ induced by $\mu$ and the Markov process $\{\eta^n_{tn^2}\; ;\; t \in [0,T]\}$. The corresponding expectation is denoted by ${{\mathbb E}}_{\mu}$.

**Notations:** Throughout the paper, for any measurable space $(U,\nu$) we denote by $L^2(\nu)$ or $L^2(U)$ the usual $L^2$–space with norm $\|\cdot\|_{L^2(\nu)}$ and scalar product $\langle \cdot,\cdot\rangle_{L^2(\nu)}$. Whenever the integration variable $u$ may be not clear to the reader, we enlighten it by denoting $L_u^2(\nu)$ or $L^2_u(U)$. We also write $f\lesssim g$ or $g \gtrsim f$ if there exists a constant $C>0$, independent of the parameters involved in $f$ and $g$, such that $f {\leqslant}Cg$. If $f \lesssim g$ and $f \gtrsim 