  In this paper, we study the edge behavior of Dyson Brownian motion with
general $\beta$. Specifically, we consider the scenario where the averaged
initial density near the edge, on the scale $\eta_*$, is lower bounded by a
square root profile. Under this assumption, we establish that the fluctuations
of extreme particles are bounded by $(\log n)^{{\rm O}(1)}n^{-2/3}$ after time
$C\sqrt{\eta_*}$. Our result improves previous edge rigidity results from
[1,24] which require both lower and upper bounds of the averaged initial
density. Additionally, combining with [24], our rigidity estimates are used to
prove that the distribution of extreme particles converges to the Tracy-Widom
$\beta$ distribution in short time.
