We consider the asymmetric simple exclusion process (ASEP) on $\mathbb{Z}$ with an initial data such that in the large time particle density $ρ(\cdot)$ a discontinuity at the origin is created, where the value of $ρ$ jumps from zero to one, but $ρ(-\varepsilon),1-ρ(\varepsilon) >0 $ for any $\varepsilon>0$. We consider the position of a particle $x_{M}$ macroscopically located at the discontinuity, and show that its limit law has a cutoff under $t^{1/2}$ scaling. Inside the discontinuity region, we show that a discrete product limit law arises, which bounds from above the limiting fluctuations of $x_{M}$ in the general ASEP, and equals them in the totally ASEP. Note: This preprint has been superseded by arXiv:1906.07711 and is no longer updated.