Convergence to the maximal-current stationary measure for half-line open TASEP

Let $\alpha \in (0,1)$ be the particle density imposed by the reservoir at site $1$, and let $\eta_t^{\mathrm{step}}$ denote the half-line open TASEP with step initial condition. For $\alpha>\frac12$, let $\mu_{1/2}^{\alpha}$ be the probability measure defined by the matrix product ansatz: choose non-negative matrices $D,E$ and vectors $\langle w|,|v\rangle$ satisfying

$$DE=\frac14(D+E),\qquad \alpha\langle w|E=\frac14\langle w|,$$ $$(D+E)|v\rangle=|v\rangle,\qquad \langle w|v\rangle=1,$$

and set

$$\mu_{1/2}^{\alpha}\{\xi\in\{0,1\}^{\mathbb N}:\xi_1=\eta_1,\ldots,\xi_L=\eta_L\}=\langle w|\prod_{x=1}^{L}\bigl(\eta_xD+(1-\eta_x)E\bigr)|v\rangle.$$

Convergence conjecture. For $\alpha>\frac12$, the half-line open TASEP with step initial condition $\eta_t^{\mathrm{step}}$ converges to the probability measure $\mu_{1/2}^{\alpha}$.

This conjecture identifies the limiting stationary measure selected by step initial data with the maximal-current stationary measure described by the matrix product ansatz. Liggett's result gives spatially correlated limiting stationary measures with asymptotic density $\frac12$ in this regime, while the asserted identification remains open and is used as a hypothesis for subsequent results in the paper.