We consider a one-dimensional system of particles, moving at constant velocities chosen independently according to a symmetric distribution on $\{-1,0,+1\}$, and annihilating upon collision -- with, in case of triple collision, a uniformly random choice of survivor among the two moving particles. When the system contains infinitely many particles, whose starting locations are given by a renewal process, a phase transition was proved to happen (see arXiv:1811.08709) as the density of static particles crosses the value $1/4$. Remarkably, this critical value, along with certain other statistics, was observed not to depend on the distribution of interdistances. In the present paper, we investigate further this universality by proving a stronger statement about a finite system of particles with fixed, but randomly shuffled, interdistances. We give two proofs, one by an induction allowing explicit computations, and one by a more direct comparison. This result entails a new nontrivial independence property that in particular gives access to the density of surviving static particles at a given time in the infinite model. Finally, in the asymmetric case, further similar independence properties are proved to keep holding, including a striking property of gamma distributed interdistances that contrasts with the general behavior.