Hydrodynamic-limit conjecture for the reshuffling stage profile

Fix $\lambda<1$. Let $\phi^N_{\mathrm{RS}}(\cdot,0)$ be the initial transformed stage profile of the reshuffling system. Suppose that it converges to a deterministic stage profile $\phi(\cdot,0)$ with bounded density. Let $\phi(\cdot,\cdot)$ denote the hydrodynamic model with initial state $\phi(\cdot,0)$.

Reshuffling hydrodynamic-limit conjecture. Under these assumptions, the process $\phi^N_{\mathrm{RS}}(\cdot,\cdot)$ converges, in an appropriate sense, to the hydrodynamic model $\phi(\cdot,\cdot)$ with initial state $\phi(\cdot,0)$.

This conjecture proposes a deterministic macroscopic evolution for the reshuffling system from general initial profiles. The source gives heuristic support and discusses a hydrodynamic model, but leaves the claimed process convergence unspecified and unproved.