Stable-regime overlap conjecture for perturbed geodesics

Let $T_0$ and $T_t$ be the passage times at perturbation times $0$ and $t$, and let $\pi_0$ and $\pi_t$ be the corresponding geodesics. Write $\operatorname{Var}(T)$ for the variance of the passage time and let $n$ tend to infinity. Stable-regime overlap conjecture. If

$$t \ll \frac{1}{n}\operatorname{Var}(T),$$

then

$$\mathbb{E}[|\pi_0 \cap \pi_t|]=\Theta(n).$$

This predicts that the geodesics retain a macroscopic expected overlap in the stable regime. The conjecture is motivated by the analogous result for Brownian last-passage percolation, while its validity in the present setting remains open.