Gaussian decay conjecture for stochastic routing probabilities

Let $h$ be the discretization parameter, let $\alpha(\boldsymbol{x})$ denote the probability of reaching the grid point $\boldsymbol{x}$ before termination under the optimal stochastic routing policy, and let $d(\boldsymbol{x})$ be the distance from $\boldsymbol{x}$ to the characteristic passing through the source point $\boldsymbol{s}$.

Gaussian decay conjecture. There exists a constant $\rho>0$ such that, for sufficiently small $h$,

$$\alpha(\boldsymbol{x})\leq e^{-\rho [d(\boldsymbol{x})]^2/h}.$$

This estimate expresses Gaussian decay away from the optimal trajectory and is intended to quantify the localization of stochastic paths near the characteristic. The supplied material does not indicate whether the claim has been proved or disproved.