Existence of the limiting distribution of the truncated extremal process

Let $\varrho_{n,n-k}$ denote the point measure of particles at generation $n$ whose most recent common ancestor with the minimum particle is at least $n-k$ generations from the root. A point measure is a measure on the real line that takes values in the nonnegative integers on bounded sets. Limiting-distribution conjecture. For every $k\in\mathbb{N}$, there exists a point measure $\varrho_k$ such that

$$\lim_{n \to \infty} \varrho_{n,n-k}=\varrho_k.$$

The authors state that they were not able to study this limiting distribution, although they believe that the law probably exists; no resolution is supplied in the source.