The even-power and independence conjecture for integrable dispersive conservation laws

Consider an integrable dispersive scalar conservation law with small parameter $\epsilon$. Its quasilinear part is the part consisting of terms linear in the derivatives of $u$; in the case $a(u)=0$, the source assumes only even powers of $\epsilon$ occur.

Even-power and independence conjecture. The quasilinear part of an integrable dispersive conservation law contains only even powers of $\epsilon$, and all coefficients of the quasilinear part are independent.

The source reports finite-order computations showing independence of all quasilinear coefficients through order $\epsilon^{12}$ and notes that the even-power restriction follows from computations at least through sixth order. The all-orders statement remains conjectural.