KPZ statistics of second class particles in ASEP via mixing

We consider the asymmetric simple exclusion process on $\mathbb{Z}$ with a single second class particle initially at the origin. The first class particles form two rarefaction fans which come together at the origin, where the large time density jumps from $0$ to $1$. We are interested in $X(t)$, the position of the second class particle at time $t$. We show that, under the KPZ $1/3$ scaling, $X(t)$ is asymptotically distributed as the difference of two independent, $\mathrm{GUE}$-distributed random variables.The key part of the proof is to show that $X(t)$ equals, up to a negligible term, the difference of a random number of holes and particles, with the randomness built up by ASEP itself. This provides a KPZ analogue to the 1994 result of Ferrari and Fontes \cite{FF94b}, where this randomness comes from the initial data and leads to Gaussian limit laws.