The second main conjecture on jeu de taquin trajectories

Fix $z\in[0,1]$ and let $(u_0,v_0)$ be the corresponding point on the Logan--Shepp--Vershik--Kerov limit curve $\Omega_*$. Let $w_1,w_2,\ldots$ be i.i.d. uniform random variables on $(0,1)$, let $T=Q_\infty(1-z,w_1,w_2,\ldots)$ be the resulting random infinite recording tableau, and let $j_n=(u_n,v_n)$ be its jeu de taquin path in the lazy parametrization. The second main conjecture. As $c\to\infty$, the random function

$$t\mapsto c^{1/4}\left[\frac{1}{\sqrt c}j_{\lceil ct^2\rceil}-(tu_0,tv_0)\right]$$

converges in distribution to a two-dimensional Brownian motion $(U_t,V_t)$ supported on the tangent line $v=\Omega_*'(u_0)u$, with

$$\mathbb E[U_tU_s]=\min(t,s)\sigma_{u_0}^2,$$

where $\sigma_{u_0}^2=\frac{\pi}{3}\sqrt{4-u_0^2}$. This is a functional refinement of the known asymptotic behavior of jeu de taquin paths and is stated as an open analogue of Donsker's theorem.