Arsie–Lorenzoni–Moro conjecture on normal forms of conservation-law deformations

Consider a deformation of the Riemann hierarchy

$$\frac{\partial u}{\partial t_d}=\partial_xP_d,\qquad P_d=\frac{u^{d+1}}{(d+1)!}+O(\varepsilon)\in\widehat{\mathcal A}_{u;0},\qquad d\ge0,$$

where the flow with respect to $t_1$ is in normal form:

$$P_1=\frac{u^2}{2}+\varepsilon a(u)v_x+\sum_{k\ge2}\varepsilon^k\sum_{\lambda\in\mathcal P_k,\,\lambda_i\ge2}c_\lambda(u)u_\lambda,$$

with $c_\lambda(u)\in\mathbb C[[u]]$. Arsie–Lorenzoni–Moro conjecture. If $a(u)=0$ and $c_2(u)\ne0$, then $c_\lambda(u)=0$ for every partition $\lambda$ with odd $|\lambda|$, so the deformation is even, and the deformation is uniquely determined by the formal power series $c_2(u),c_4(u),\ldots$. The conjecture asserts a normal-form classification by the even coefficients; the source gives no resolution.