Equilibrium measure conjecture for TASEP on Galton–Watson trees

Consider TASEP with a reservoir of rate $\lambda=\rho q$ for some $\rho \in (0,\infty)$, and suppose that a flow rule holds for a flow of strength $q$. Let $\pi_{\lambda}$ be the equilibrium measure obtained as in the convergence from the all-empty configuration. For $\rho\leq \frac{1}{2}$, the equilibrium measure conjecture.

$$\pi_{\lambda}=\nu_{\rho}.$$

For $\rho>\frac{1}{2}$, it should satisfy

$$\lim_{\lvert x\rvert \rightarrow \infty}\pi_{\lambda}(\eta(x)=1)=\frac{1}{2}.$$

This conjecture describes the expected equilibrium behaviour of the TASEP on a Galton–Watson tree, in analogy with the TASEP on the half-line. The supplied text presents it as the paper's final open problem; no resolution is given.