a natural *height function* $h(x)$ associated to the occupation variable $\eta(x)$, and in particular its increments satisfy: $$h(x+1)-h(x) = \eta(x) -\rho.$$ Because of the presence of the reservoirs in our dynamics, the microscopic height process $\{h_t(x)\; ; \; t {\geqslant}0, x=1,\dots,n\}$, which is derived from the Markov process $\eta_t$ defined above, has to be carefully defined. We refer the reader to Section \[ssec:height\] for a rigorous definition. Similarly to , we are interested in the macroscopic limit of height fluctuations starting the evolution from $\nu_\rho^n$ and looking in the time scale $tn^2$. In this case the averaged local height at site $x \in \{1,\dots,n\}$ and microscopic time $tn^2$ is equal to $c_n t$, where $c_n$ is related to the initial density $\rho$ and the strength of the asymmetry $\varepsilon_n$, as follows: $c_n=n^2\varepsilon_n\rho(1-\rho)t$. Therefore, the *microscopic height fluctuation field* is given by $$\label{eq:height-field-intro}
{{\mathcal Z}}_t^n(\cdot) = \frac{1}{n^{3/2}} \sum_{x=1}^n \big(h_{tn^2}(x)-c_n t\big)\delta_{x/n}(\cdot),$$ which means, formally, that ${{\mathcal Y}}_t^n = -\nabla {{\mathcal Z}}_t^n$. In the same spirit as Theorem \[theo:convergence\], for $\varepsilon_n=E/\sqrt n$ and $\rho=\frac12$, we prove in Theorem \[theo:convheight\] that the sequence of processes ${{\mathcal Z}}^n$ converges, in a suitable space of distributions, towards the unique energy solution ${{\mathcal Z}}$ to with Neumann 