Level classification conjecture for scalar Poisson lambda-brackets

Let a non-quasiconstant coefficient Poisson $\lambda$-bracket on $\widetilde{\mathcal{R}}_1$ have odd order $N$ and level $m$, where the level is defined by $m=\max_j\{j+\operatorname{ord}(f_j)\}$ for $\{u_\lambda u\}=\sum_{j\text{ odd}}(D+2\lambda)^j f_j$. Level classification conjecture. With the exception of level $m=1$ in the case $N=3$, the only possible values of the level are $m=N$, $N+1$, or $N+2$; all these values occur, except $m=2$ when $N=1$. This is motivated by classifications of Hamiltonian operators through order five, while the general classification remains open.