Canonical-form conjecture for tau-symmetric deformations of the Riemann hierarchy

Let a tau-symmetric integrable Hamiltonian deformation of the Riemann hierarchy be considered modulo normal Miura-type transformations. Write $w_k=\partial_x^k w$. The canonical deformation is required to have

$$\frac{\partial w}{\partial t^q}=\frac{\partial}{\partial x}\left(\frac{\delta H_q}{\delta w(x)}\right),\qquad q\geq 0,$$

and its Hamiltonian density $h_1$ has the displayed standard form through order $\epsilon^{12}$, with coefficients $a_0,a_i,b_i$. Starting from $\epsilon^4$, the displayed terms are selected so that the factor with highest derivative in each monomial is nonlinear and no term contains a $w_x$ factor; if $a_0=0$, all $a_j,b_j$ vanish, while if $a_0\ne0$, the coefficients $b_j$ are uniquely determined by $a_0,a_1,a_2,\dots$. Canonical-form conjecture. Any tau-symmetric integrable Hamiltonian deformation of the Riemann hierarchy is equivalent, under a normal Miura-type transformation, to this canonical tau-symmetric integrable deformation, which is uniquely determined by the stated standard form of $h_1$. The source presents this as a conjecture and provides no general proof; only finite-order equivalences are reported elsewhere in the conclusion.