GUE Tracy–Widom fluctuation conjecture for individual sticky Brownian atoms

For fixed $x$, let $\gamma_t(tx,dw)$ denote the law of the relevant atomic mass and let $X_{x,t}$ be a random variable on $\mathbb{R}$ with law

$$\sqrt{2\pi t}\,e^{-t x^2/2}w\,\gamma_t(tx,dw).$$

GUE Tracy–Widom fluctuation conjecture. There are functions $J:\mathbb{R}\to\mathbb{R}$ and $\sigma:\mathbb{R}\to\mathbb{R}$ such that

$$\lim_{t\to\infty}\mathbb{P}\left(\frac{\log(X_{x,t})+J(x)t}{t^{1/3}\sigma(x)}<z\right)=F_{GUE}(z),$$

where $F_{GUE}$ is the cumulative distribution function of the GUE Tracy–Widom distribution. The conjecture predicts $t^{1/3}$-scale GUE fluctuations for individual atoms in the large-time regime; the source does not provide explicit formulas for $J$ or $\sigma$, nor state a resolution.