Scaling-limit conjecture for ancestral lineages in ranked tree-child networks

Let $X^{(\ell)}_k$ denote the number of lineages in the ancestry of a leaf after moving $k$ steps away from that leaf in a uniformly sampled ranked tree-child network with $\ell$ leaves. For fixed $M$, consider the rescaled random variable at time $\lfloor \ell-M\sqrt{\ell}\rfloor$. Scaling-limit conjecture. The sequence of random variables

$$\frac{1}{\sqrt{\ell}}X^{(\ell)}_{\lfloor\ell-M\sqrt{\ell}\rfloor}$$

converges in distribution to a positive random variable $W_M$. This conjecture gives a precise version of the paper's broader prediction that the lineage-counting Markov chain, suitably rescaled as the number of leaves tends to infinity, converges to a smooth function with a random shape parameter. The preceding proposition supplies expectation bounds, but convergence in distribution is left conjectural.