Conjecture on extremal stationary measures for half-space KPZ

Let $\mathcal{H}_{u,v}$ denote the explicitly constructed stationary processes for the half-space KPZ equation with Neumann boundary parameter $u$, indexed by $v\leqslant\min\lbrace 0,u\rbrace$. Extremal stationary-measure conjecture. The family $\{\mathcal{H}_{u,v}\}_{v\leqslant\min\lbrace 0,u\rbrace}$ constitutes all extremal stationary measures for the half-space KPZ equation. Moreover, for initial data with drift $-v$ at infinity, the centered solution converges to the corresponding member of this family according to the phase-dependent alternatives stated in the conjecture: for $u\geqslant0$, the limit is $\mathcal{H}_{u,v}$ when $v\leqslant0$ and $\mathcal{H}_{u,0}$ when $v\geqslant0$; for $u\leqslant0$, it is $\mathcal{H}_{u,v}$ when $v\leqslant u$ and $\mathcal{H}_{u,u}$ when $v\geqslant u$. This conjecture proposes a complete classification of extremal stationary measures and describes the expected long-time phase diagram for general initial data.