Geodesic--argmax rigidity conjecture for the directed landscape

Let $\mathcal L$ be the directed landscape, let $\pi^*(\tau T)$ denote the spatial location at time $\tau T$ of the relevant geodesic, and condition on $\mathcal L(0,0;0,T)=L$. For each $\tau\in(0,1)$, define the rescaled spatial argument of the directed-landscape argmax by

$$\operatornamewithlimits{argmax}_{x\in\mathbb R}\mathcal L\left(0,0;\frac{xT^{3/4}}{2L^{1/4}},\tau T\right).$$

Geodesic--argmax rigidity conjecture. For every $\tau\in(0,1)$ and $T>0$, conditionally given $\mathcal L(0,0;0,T)=L$, as $L\to\infty$,

$$\frac{2L^{1/4}\pi^*(\tau T)}{T^{3/4}}-\operatornamewithlimits{argmax}_{x\in\mathbb R}\mathcal L\left(0,0;\frac{xT^{3/4}}{2L^{1/4}},\tau T\right)\to 0$$

in probability. The conjecture asserts that the geodesic location and the argmax process become asymptotically identical under the high-height conditioning. It is motivated by the more involved definition of the geodesic and by the expected uniqueness of the conditional argmax process.