Strong convergence conjecture for fully nonlinear KdV-type equations

Consider the scalar conservation law

$$u_t+f(u)_x=0.$$

Let $u$ solve the fully nonlinear fourth-order approximation

$$u_t+f(u)_x-b4g(u_{xx})_x=-b5u_{xxxx},\qquad b4,b5>0,$$

where $f$ is strictly convex, $g$ is concave, and $g(0)=0$. Strong convergence conjecture. The solution strongly converges to the unique entropy solution of the scalar conservation law as $b4$ and $b5b4^{-1}$ tend to $0$. The conjecture is motivated by numerical evidence showing a single dispersive ripple that vanishes as $b4\to0$, in contrast with the growing dispersive wave trains of the KdV equation. A rigorous theory for these fully nonlinear evolution equations remains challenging, and the source presents the claim as a conjecture rather than a proved result.