The stability-to-noise-sensitivity transition conjecture for geometric last-passage percolation

Consider planar last-passage percolation with geometric weights. Let $T_n(\omega)$ and $T_n(\tilde\omega^t)$ denote the last-passage times in the original and site-resampled environments, respectively, where $t$ is the resampling parameter. Stability-to-noise-sensitivity transition conjecture. The transition from stability to noise sensitivity occurs at $t\asymp n^{-1/3}$ in the sense that, as $n\to\infty$, (i) for $t\ll n^{-1/3}$,

$$\operatorname{Corr}\big(T_n(\omega),T_n(\tilde\omega^t)\big)=1-o(1);$$

and (ii) for $t\gg n^{-1/3}$,

$$\operatorname{Corr}\big(T_n(\omega),T_n(\tilde\omega^t)\big)=o(1).$$

This conjecture predicts the KPZ-scale transition between stability and noise sensitivity for geometric last-passage percolation; the supplied context gives evidence for the scale but does not establish the stated correlation behavior in both regimes.