Conjecture on reducing Hamiltonian operators to the Gardner operator

Let $H$ be a translation-invariant Hamiltonian operator of order $N\geq 7$. The Gardner operator is the first-order operator $D$.

Gardner normal-form conjecture. $H$ can be transformed into the Gardner operator $D$ by either a transformation from the potentiation normal-form conjecture or a composition thereof with a transformation

$$x=y,\qquad v=\sum_{i=0}^k a_i(x)v_i.$$

This is the strongest conjecture stated in the paper. It follows the lemma asserting that a quasiconstant skew-adjoint differential operator can be reduced to the Gardner operator by a transformation of the displayed form; no resolution is supplied.