Conjecture on finite-population traveling waves and propagation speed

Let $mu_0^K$ be a compactly supported initial distribution, let $fxi^K(t)=H_K(fu^K_t)$ denote the position of the particle of rank $K$, and let $\hat{\mu}^{K}_t:=\mu^K_t(\cdot-fxi^K(t))$ be the population distribution in the moving frame. Write

$$\sigma^*=\left\{\begin{array}{ll}\chi+\frac{1}{\chi} & \text{if }\chi>1,\\2 & \text{if }\chi\leq 1.\end{array}\right.$$

Finite-population traveling-wave conjecture. There exists a propagation speed $\sigma^K$ such that, in an appropriate convergence sense, $\lim_{t\to+\infty}\frac{\xi^K(t)}{t}=\sigma^K$; for every compact subset $\mathcal K\subset\mathbb R$, $\hat{\mu}^{K}_t$ converges in law on $\mathcal K$ to a stationary distribution $\hat{\mu}^{K}_{\infty}$; and $\lim_{K\to+\infty}\sigma^K=\sigma^*$.

These assertions formalize the numerically observed linear spreading, local convergence to a stationary traveling-wave profile in the moving frame, and convergence of the finite-population speed to the deterministic speed. The source presents them as open problems, and the precise meaning of the convergence in the first assertion remains unspecified.