Mokhov's conjecture on the Doyle–Potemin form of local Poisson operators

Let $P=\sum_{e=1}^{d+2} P^{ij}_e {\partial}_x^{d+2-e}$ be a local operator of homogeneous differential order $d+2$, meaning that

$$\deg_{{\partial}_x} P^{ij}_e=e,$$

where $d\geq 0$. Assume that $P$ defines a Poisson bracket. Mokhov's conjecture. There exists a local skew-symmetric operator $Q^{ij}$ of homogeneous differential order $d$ such that

$$P={\partial}_x\circ Q^{ij}\circ {\partial}_x.$$

This conjecture concerns the possible form of homogeneous local Poisson structures of differential degree at least two. Doyle and Potëmin proved the asserted form for differential degrees $2$ and $3$, while the general case remains open in the source.