Convergence conjecture for kinetic functions of high-order equivalent equations

Consider a scalar conservation law

$$\partial_t u+\partial_x f(u)=0,$$

where the flux function $f$ admits a single inflection point. Let $\varphi^\flat_\alpha$ be the exact kinetic function associated with the model, and let $\psi^{\flat,q}_\alpha$ be the kinetic function associated with a numerical scheme whose equivalent equation has the form specified in equation (EQ.7), with $q$ denoting the order of accuracy of the scheme.

Convergence conjecture. As $q\to\infty$, the kinetic function $\psi^{\flat,q}_\alpha$ converges to the exact kinetic function $\varphi^\flat_\alpha$:

$$\lim_{q\to\infty}\psi^{\flat,q}_\alpha=\varphi^\flat_\alpha.$$

The conjecture expresses the expectation that increasingly accurate equivalent equations capture the kinetic function of the continuous model, despite the observed discrepancy between kinetic functions generated by finite-difference schemes and the exact kinetic function. The source provides no resolution, so the conjecture remains open.