Bolthausen–Sznitman scaling-limit conjecture for the ancestral partition process

For each fixed sample size $k\ge1$, let $(\Pi_t^{k,N},t\ge0)$ be the ancestral partition process of a population of size $N$, restricted to the sample, and let $(\Pi_t^k,t\ge0)$ denote the restriction to $[k]$ of a Bolthausen–Sznitman coalescent. Bolthausen–Sznitman scaling-limit conjecture. After speeding up time by a factor $(\log N)^3$, for every $k\ge1$,

$$(\Pi_{t(\log N)^3}^{k,N},t\ge0)\overset{d}{\to}(\Pi_t^k,t\ge0)$$

in the sense of finite-dimensional distributions. This predicts that the genealogy of a fixed sample converges to the Bolthausen–Sznitman coalescent on the time scale $(\log N)^3$; the source describes the supporting arguments as convincing but not fully rigorous.