by a multi-scale argument (also given in [@G]), which consists in replacing, at each step, a local function of the dynamics by another function whose support increases and whose variance decreases, and its proof used a key spectral gap inequality which is uniform in the density of particles, and is not available for many models. Later in [@GJS] it is assumed that the models satisfy a spectral gap bound which does not need to be uniform in the density of particles and allows more general jump rates. More recently in [@fgsimon2015; @GJSi], and then in [@blondel2016], a new proof of the BGP has permitted to extend the previous results to models which do not need to satisfy a spectral gap bound, as, for example, symmetric exclusion processes with a slow bond, and microscopic dynamics that allow degenerate exchange rates. In this paper, we adapt that strategy, which turns out to be quite robust, to our finite model with stochastic boundary reservoirs. This is the goal of Theorem \[theo:BG\].

### Boundary behavior at the microscopic scale

As we already mentioned, the KPZ equation and SBE are closely related, and this relation can be seen also at the microscopic level. There is 