The starting point of a directed geodesic is special

For any $t\in [0,1\mathrm{/}2)$, consider the process ${\eta }_t:[0,1\mathrm{/}2]\mapsto \mathbb{R}$ defined by ${\eta }_t(s)=\mathrm{\Pi }(t+s)-\mathrm{\Pi }(t)$, where $\mathrm{\Pi }$ is the directed geodesic from $(0,0)$ to $(0,1)$ in the directed landscape. Let $0\le t<u<1\mathrm{/}2$. We show that the laws of ${\eta }_t$ and ${\eta }_u$ are mutually absolutely continuous if and only if $t>0$. This proves Conjecture 14.5 in Dauvergne, Ortmann, and Virág, 2022 in the affirmative.