The statistical description of the scalar conservation law of the form $\rho_t=(H(\rho))_x$ with $H: \mathbb{R} \to \mathbb{R}$ a smooth convex function has been an object of interest when the initial profile $\rho(\cdot,0)$ is random. The special case when $H(\rho)=\frac{\rho^2}{2}$ (Burgers' equation) has in particular received extensive interest in the past and is now understood for various...