The starting point of a directed geodesic is special

For any $t\in [0,1/2)$, consider the process $η_t:[0,1/2]\mapsto\mathbb{R}$ defined by $η_t(s)=Π(t+s)-Π(t)$, where $Π$ is the directed geodesic from $(0,0)$ to $(0,1)$ in the directed landscape. Let $0\leq t<u<1/2$. We show that the laws of $η_t$ and $η_u$ are mutually absolutely continuous if and o