GOE fluctuation conjecture for flat initial conditions

Let $P$ be the pressure function, let $\beta$ be the slope of flat initial conditions, and set $\lambda=\gamma P”(\beta)\neq 0$ and $A=-P'(\beta)$. For the particle system with flat initial data of slope $\beta$, define $F_{\mathrm{GOE}}(s)=\mathbb{P}(\xi_{\mathrm{GOE}}\leq s)$. GOE fluctuation conjecture. For each $u$ and $s$,

$$\lim_{t\to\infty}\mathbb{P}\left(x_{\lfloor ut\rfloor}(t)-(u\beta+\gamma P(\beta))t\leq-\operatorname{sgn}(\lambda)(|\lambda|A^2t)^{1/3}s\right)=F_{\mathrm{GOE}}(2s).$$

The conjecture predicts Tracy–Widom GOE fluctuations, in contrast with the GUE law expected for wedge initial conditions; the corresponding spatial fluctuation process is expected to be the Airy$_1$ process. The statement is motivated by the KPZ equation and periodic TASEP, where analogous formulas are known.