Hydrodynamic limit for the boundary driven zero-range process

Let $I_N$ be the lattice site set, let $\Omega_N$ be the configuration space, and let $\{\mu^N\}_{N\in\mathbb N}$ be probability measures on $\Omega_N$ satisfying $\mu^N\leq\bar\nu^N$. Suppose that $\{\mu^N\}$ is associated with a continuous profile $\gamma:[0,1]\to\mathbb R_+$, and let $\mathbb P_{\mu^N}$ denote the law of the process with generator $N^2L_N$. For each $t\in[0,T]$, let $G:[0,1]\to\mathbb R$ be continuous and let $\delta>0$. Hydrodynamic limit. Then

$$\lim_{N\to+\infty}\mathbb P_{\mu^N}\left[\left|\frac{1}{N}\sum_{x\in I_N}G\left(\frac{x}{N}\right)\eta_t(x)-\int_0^1G(u)\rho_t(u)\,du\right|>\delta\right]=0,$$

where, for $\theta=1$, $\rho_t(u)$ is a weak solution of the hydrodynamic equation with Robin boundary condition ($\kappa=1$), while, for $\theta>1$, it is a weak solution with Neumann boundary condition ($\kappa=0$). The claim is the expected hydrodynamic limit for the process; the source explicitly says that parts of the proof were not yet completed, so the result remains conjectural.