Distributional equivalence conjecture for physical and non-physical particle counts

Let $N\geq 1$, let $\tilde{X}_N$ be the random number of particles remaining in the physical perfectly plastic collision process, and let $\tilde{Z}_N$ be the corresponding random variable in the non-physical model. Distributional equivalence conjecture. For $N\geq 1$, $\tilde{X}_N$ and $\tilde{Z}_N$ have the same distribution. Equivalently, the distribution, mean, and variance formulas established for $\tilde{Z}_N$ should remain valid after replacing $\tilde{Z}_N$ by $\tilde{X}_N$. The distributions coincide exactly for $N=2$ and $N=3$, and simulations provide evidence for larger $N$; the assertion for all $N\geq 1$ remains open.