The quasilinear-part conjecture for viscous conservation laws

Consider a scalar conservation law with viscosity, written in the form

$$u_t=\partial_x\left[u^2+\epsilon a(u)u_x+\epsilon^2b_1(u)u_{xx}+\epsilon^3c_1(u)u_{xxx}+\epsilon^4d_1(u)u_{4x}+\epsilon^5e_1(u)u_{5x}+\mathcal{O}(\epsilon^6)+\cdots\right],$$

where $a(u)$ is the coefficient of the first-order viscous term. The function $a(u)$ is called the viscous central invariant.

Quasilinear-part conjecture. The quasilinear part of a viscous conservation law is uniquely determined by $a(u)$.

The source motivates this conjecture from a classification theorem valid through finite order in $\epsilon$, where the higher displayed coefficients are determined by $a(u)$. It does not state that the all-orders claim has been proved.