The directed-landscape geodesic upper-tail rate conjecture

Let $pi$ be the $L$-geodesic from $(0,0)$ to $(0,1)$. For $t i (0,1/2]$, define

$$\iota(t)=\frac{-(2t)^{5/2} (9 b+4)+6 t^2 (25 b+13)-2(2 t)^{3/2} (26 b+19)-48 \sqrt{2t}+24}{3 \left(3-\sqrt{8t}\right)^3 (1-t)^2 t^2},$$

where

$$b=\frac{\sqrt{72 t^2+6 (2t)^{3/2}-143 t-12 \sqrt{2t}+72}}{(9-8 t) \sqrt{t}}.$$

The directed-landscape geodesic upper-tail rate conjecture. For $t\in (0,1/2]$, as $a\to\infty$,

$$P(\pi(t)\ge a)=e^{-\iota(t) a^3+o(a^3)}.$$

In particular, as $t\to 0$,

$$\iota(t)=\frac8{27}/t^{2}+o(1/t^2).$$

This conjecture gives the proposed large-deviation rate for a geodesic forced to lie high at time $t$; the paper notes that the corresponding heuristic path is not generally piecewise linear when $t\ne1/2$, but instead involves two linear pieces and a parabola. The optimization leading to the formula is beyond the scope of the paper, and no resolution is supplied.