Uniform particle-distance bound for the Newtonian shock-wave model

Let $W\in \mathcal{C}^2(\mathbb{R})$ satisfy the convexity condition $\mathrm{Convex}$. Let $\phi_0^x$ and $\phi_0^\tau$ belong to $\mathcal{C}^1_{p,x}$. For the particle positions $X_j(t)$ solving the Newton equation with the initial and boundary conditions $\mathrm{IDD}$ and $\mathrm{BCD}$, write $U_j^N$ for the distance between neighboring particles.

Uniform particle-distance conjecture. There exist $T>0$ and $a<b$, depending only on $W$, $\phi_0^x$, and $\phi_0^\tau$, such that

$$\na\leq U_j^N(N\tau)\leq b,\qquad \forall j\in[-N,N-1],\quad \forall N\in\mathbb{N},\quad \forall \tau\in[0,T].$$

The conjecture would provide the uniform boundedness of the interparticle distances required for the paper's nonlinear results. The authors state that they have not been able to prove this assumption.