Conjecture on stationary distributions of the centered particle system

Let $f^n(\cdot,t)$ denote the empirical distribution of the locations of an $n$-particle stochastic system, and let $f^n_*(\cdot,t)$ be its recentered version whose median is at $0$. Assume the finite-moment condition and the positive-density condition on the jump-size distribution. Let $f^n_*(\cdot,\infty)$ be a random value of $f^n_*(\cdot,t)$ in the stationary regime, and let $\phi(\cdot)$ be the unique traveling-wave shape, centered to have median at $0$. Stationary-distribution convergence conjecture. As $n\to\infty$, $f^n_*(\cdot,\infty)$ concentrates at $\phi(\cdot)$, namely

$$\\|f^n_*(\cdot,\infty)-\phi(\cdot)\\|\stackrel{P}{\rightarrow}0.$$

For each fixed $n$, the recentered particle system is stated to have a unique stationary distribution, while the mean-field model has a unique traveling shape up to translation. The conjecture asserts that the stationary finite-particle system converges to this deterministic traveling shape in the large-population limit; this is suggested by the convergence from the particle system to the mean-field model and from the mean-field model to the traveling wave.