The hydrodynamic-limit conjecture for ASEP with time-dependent reservoirs

Let $u$ solve the hydrodynamic equation

$$\partial_tu+p\partial_x\mathcal Q(u,v)=0$$

with boundary parameters determined by the reservoir rates $(\alpha,\beta,\gamma,\delta)(t)$ through the relations defining $\nu_\pm(t)$. Hydrodynamic-limit conjecture. For the case $\theta=0$, the hydrodynamic equation is given by the equation with $\nu_\pm=\nu_\pm(t)$ determined through the boundary relations with $(\alpha,\beta,\gamma,\delta)=(\alpha,\beta,\gamma,\delta)(t)$. This conjecture extends the hydrodynamic description to time-dependent reservoir parameters; the supplied text does not state whether it has been proved or remains open.