Uniform-in-time convergence conjecture for the nondiffusive advection scheme

Let $u^0$ be a function with total bounded variation, and associate to it the initialization

$$u_j^0=\frac{1}{\Delta x}\int_{x_{j-1/2}}^{x_{j+1/2}}u^0(x)\,dx.$$

Suppose that the ratio $\lambda=\frac{\Delta t}{V\Delta x}$ is fixed and belongs to $(0,1]\setminus\{1/2\}$. Uniform-in-time convergence conjecture. There exists a constant $C>0$, depending only on $u^0$ and $\lambda$, such that

$$\forall n\in\mathbf{N},\qquad \Delta x\sum_{j\in\mathbf{Z}}\left|u_j^n-\frac{1}{\Delta x}\int_{x_j-\Delta x/2}^{x_j+\Delta x/2}u(n\Delta t,x)\,dx\right|\leq C\sqrt{\Delta x}.$$

The conjecture expresses that the scheme's plateau-forming behavior acts as a global attractor and yields a time-independent error bound for bounded-variation initial data. The source gives numerical evidence but no proof or resolution.