Small deviation conjecture for geodesics in semi-discrete Brownian last passage percolation

Let $\{B_i(\cdot)\}_{i\in\mathbb{Z}}$ be a sequence of two-sided standard Brownian motions on $\mathbb{R}$, and let $\Pi_n$ be the almost surely unique geodesic from $(0,0)$ to $(n,n)$ in semi-discrete Brownian last passage percolation. For $s\in(0,1)$, write $\Pi_n(ns)$ for the spatial coordinate of the geodesic at longitudinal parameter $ns$. Brownian geodesic small deviation conjecture. There exists $\delta_0>0$ and positive constants $C_1,c_1,C_2,c_2$ such that, for all $0<\delta<\delta_0$ and all sufficiently large $n$ depending on $\delta$,

$$C_2e^{-c_2\delta^{-3/2}}\leq \mathbb{P}\left(\sup_{s\in(0,1)}n^{-2/3}|\Pi_n(ns)-ns|\leq\delta\right)\leq C_1e^{-c_1\delta^{-3/2}}.$$

This is proposed as the Brownian last-passage-percolation analogue of the corresponding small-deviation estimate, motivated by the curvature of the limit shape and one-point moderate-deviation estimates; the parallelogram estimates needed for a proof had not been established in the cited literature.