Hydrodynamic limit conjecture for the BHP

Let $\rho_0: \mathbb{R}\rightarrow[0,1]$ be an initial density profile, and let $\{\nu^{\epsilon}\}_{\epsilon\in\mathbb{R}_{>0}}$ be probability measures on the configuration space $\mathcal{C}$ associated to $\rho_0$. For fixed $\epsilon>0$, let $\eta_t^{\epsilon}\in\mathcal{C}$ be the BHP configuration at time $t>0$ under time acceleration by $\epsilon^{-1}$, with initial law $\nu^{\epsilon}$. Let $\pi_t^{\epsilon}$ be the corresponding random empirical measure, and write $\pi_t(dz)=\rho(z,t)\,dz$. The BHP hydrodynamic-limit conjecture. For every $t>0$, $\pi_t^{\epsilon}$ converges in probability to $\pi_t$ in the sense of the stated empirical-measure convergence, where $\rho$ solves

$$\frac{\partial\rho(t,z)}{\partial t}=\frac{\partial}{\partial z}\left[\frac{1-\rho(t,z)}{\rho(t,z)}\int_z^{\infty}\rho(t,w)\,dw\right],\qquad \rho(0,z)=\rho_0(z).$$

This conjecture proposes the hydrodynamic limit for the BHP, analogous to the known hydrodynamic limit for TASEP, but with a nonlocal evolution equation determined by the BHP generator. The source provides no resolution, so the status remains open.