GOE Dyson Brownian motion conjecture for the Airy one process

Let $\lambda_{\max}^{(N)}(t)$ be the largest eigenvalue of an $N\times N$ matrix undergoing Dyson's Brownian motion with parameter $\beta=1$. Let ${\cal A}_{\rm 1}$ denote the Airy one process. GOE Dyson Brownian motion conjecture. The evolution of the largest eigenvalue of $N\times N$ matrices for $\beta=1$ Dyson's Brownian Motion converges, in the limit $N\to\infty$ and properly rescaled, to the process ${\cal A}_{\rm 1}$. The conjecture is motivated by the established correspondence between GOE one-point statistics and the top layers of the multilayer flat PNG model, while the full process-level correspondence remains open.