Diffusive-scaling conjecture for the detachment-counting process

Consider the detachment process coupling all $n$-detachment processes on one probability space. Let $\hat\tau^{(n)}$, for $n=2,3,\ldots$, be the corresponding first detachment times, and define, for $k\ge 1$,

$$\mathcal{M}(k):=\max\{n:\ \hat\tau^{(n)}\le k\},$$

with $\mathcal{M}(k):=1$ when no two passengers have been separated. Extend $\mathcal{M}$ to all $t\ge 1$ by linear interpolation and set

$$\mathcal{M}^{(m)}(t):=\frac{\mathcal{M}(m^2t)}{m}.$$

Diffusive-scaling conjecture. The laws of the processes $\mathcal{M}^{(m)}$, as $m\to\infty$, have a limit. The process has monotone non-decreasing paths, and the conjecture asks for a process-level limit under diffusive time scaling and linear spatial scaling. The source does not specify the mode of convergence or identify the limiting law.