Conjecture on reducing Hamiltonian operators by contact transformations and potentiation

Let $H$ be a translation-invariant Hamiltonian operator of order $N\geq 7$. A potentiation is the transformation

$$x=y,\qquad u=v_y.$$

A quasiconstant coefficient skew-adjoint differential operator has coefficients depending only on $x$.

Potentiation normal-form conjecture. $H$ can be transformed into a quasiconstant coefficient skew-adjoint differential operator using either a contact transformation or a composition thereof with the potentiation.

This is presented as a stronger reformulation of the preceding conjecture, using the extended transformation $x=v$, $u=1/v_y$ and its decomposition into potentiation and hodograph transformations. Its resolution is not given in the source.