Normal-form classification conjecture for integrable RH conservation laws

Consider an integrable hierarchy of scalar conservation laws

$$\frac{\partial u}{\partial t_n}=\partial_x\omega_n,$$

which is a perturbation of the Riemann--Hopf hierarchy, and transform it to the Arsie--Lorenzoni--Moro normal form

$$\widetilde{\omega}_1=\frac{w^2}{2}+a(w)w_x+\sum_{k\geq2}\epsilon^k\sum_{\substack{\lambda\in\mathcal{P}_k\m_1(\lambda)=0}}c_\lambda(w)w_\lambda.$$

Assume $a(w)=0$ and $c_2(w)\neq0$. Normal-form classification conjecture. All coefficients $c_\lambda(w)$ with odd $|\lambda|$ vanish; the functions $c_2(w),c_4(w),\ldots$ uniquely determine every other $c_\lambda(w)$; and $c_2(w),c_4(w),\ldots$ are arbitrary free functional parameters. The source supplies no evidence of resolution for this stated claim.