Relaxation-time invariance of post-collision amplitude and downstream density

Let $\tau$ be the relaxation time, and consider the jamiton produced by a collision. Denote its amplitude by $A$ and its downstream density by $\rho_+$. Amplitude and downstream-density invariance conjecture. The amplitude and value of $\rho_+$ of the jamiton resulting from a collision are independent of $\tau$. The claim is motivated by numerical experiments and by the observation that $\rho_+$ and $\rho_-$ depend only on the sonic density, which is reported to be independent of $\tau$; it remains unproved.