The Plancherel-TASEP second-class-particle Brownian-motion conjecture

Consider the Plancherel-TASEP interacting particle system, and let $u_n$ be the position of its second-class particle at time $n$. Let $\mu_{\operatorname{SC}}$ be the semicircle distribution on $[-2,2]$, and let $\sigma_V$ denote the scaling constant used in the conjecture. Plancherel-TASEP second-class-particle conjecture. There exists a random variable $V\sim\mu_{\operatorname{SC}}$ such that, as $c\to\infty$,

$$t\mapsto\frac{c^{1/4}}{\sigma_V}\left[\frac{u_{\lfloor ct^2\rfloor}}{\sqrt c}-Vt\right]$$

converges in distribution to the standard Brownian motion $B(t)$. This reformulates part of the jeu de taquin conjecture in interacting-particle language, identifying the second-class particle's fluctuations with Brownian motion.