Path-space hydrodynamic limit for the boundary driven zero-range process

Let $Q^N$ be the law on $\mathcal D([0,T],\mathcal M_+)$ of the empirical process $\{\pi_t^N:t\in[0,T]\}$. Let $\rho:[0,T]\times[0,1]\to\mathbb R_+$ be a weak solution of the hydrodynamic equation with initial profile $\gamma$, with boundary parameter $\kappa=1$ when $\theta=1$ and $\kappa=0$ when $\theta>1$. Hydrodynamic limit. As $N\to\infty$, $Q^N$ converges weakly to the probability measure $Q$ concentrated on the trajectory

$$\pi_t(du)=\rho_t(u)\,du,$$

where $\rho_t(u)$ is the hydrodynamic profile. This is the path-space formulation of the preceding hydrodynamic-limit claim, identifying the limiting empirical trajectory with the solution of the hydrodynamic equation. The source presents it as conjectural because the proof of the hydrodynamic limit is not complete.