Absolute continuity conjecture for directed-geodesic increments

Let $\Pi=\Pi_{0,0}$ be the directed geodesic from $(0,0)$, and define the increment process

$$\eta_t:[0,1/2]\to\mathbb{R},\qquad \eta_t(s)=\Pi(t+s)-\Pi(t).$$

For $0\le t<u<1/2$, absolute continuity conjecture. The laws of $\eta_t$ and $\eta_u$ are mutually absolutely continuous if and only if $t>0$. The source presents this as a conjectural distinction between the beginning of a directed geodesic and later increments, but gives no proof or resolution.