One Force -- One Solution principle for the open KPZ equation

Fix $u,v\in\R$. Let $h_0,\tilde h_0\in\operatorname{C}([0,1])$ be arbitrary random functions on the same probability space, and let $\xi$ be white noise on $\mathbb{R}\times[0,1]$. Let $h(t,x)=h(t,x;t_0)$ and $\tilde h(t,x)=\tilde h(t,x;t_0)$ solve the open KPZ equation driven by $\xi$, started at time $t_0\leq 0$ from $h_0$ and $\tilde h_0$, respectively. One Force -- One Solution principle. For every real $s<s'$, the random functions $(t,x)\mapsto h(t,x)-h(t,0)$ and $(t,x)\mapsto\tilde h(t,x)-\tilde h(t,0)$ converge almost surely in $\operatorname{C}([s,s'];\widetilde{\operatorname{C}}([0,1]))$ to the same limit as $t_0\to-\infty$. This principle is stated as a conjecture for the open KPZ equation; the supplied context gives no resolution status.