The first main conjecture on fluctuations of Schensted row insertion

Fix $z\in[0,1]$. Let $(u_0,v_0)=(\operatorname{RSK\ cos}z,\operatorname{RSK\ sin}z)$ be the corresponding point on the Logan--Shepp--Vershik--Kerov limit curve $\Omega_*$. For i.i.d. random variables $w_1,w_2,\ldots\sim U(0,1)$, let $(x_n,y_n)=\operatorname{Ins}(w_1,\ldots,w_n;z)$ be the position of the box created by inserting $z$. The first main conjecture. The random points

$$n^{1/4}\left[\left(\frac{x_n-y_n}{\sqrt n},\frac{x_n+y_n}{\sqrt n}\right)-(u_0,v_0)\right]$$

converge in distribution to a centered degenerate Gaussian distribution supported on the line $v=\Omega_*'(u_0)u$. This distribution is determined by

$$n^{1/4}\left[\frac{x_n-y_n}{\sqrt n}-u_0\right]\xrightarrow[n\to\infty]{\mathcal D}N(0,\sigma_{u_0}^2),$$

where

$$\sigma_{u_0}^2=\frac{\pi}{3}\sqrt{4-u_0^2}.$$

This refines the known law of large numbers for the insertion position and predicts fourth-root Gaussian fluctuations tangent to the limit curve; the claim is presented as an unproved main conjecture.