Polynomiality conjecture for compatible Ferapontov and Doyle--Potëmín operators

Let $P$ be a Ferapontov-type operator parameterised by a metric $g_{ij}$, and let $R$ be a Doyle--Potëmín operator parameterised by a Monge metric $f_{ij}$. Suppose the Monge decomposition is

$$f_{ij}=\psi^\alpha_i\phi_{\alpha\beta}\psi_j^\beta.$$

For the inverse metric $g^{ij}$, define the symmetric matrix

$$Q^{\alpha\beta}=\psi^\alpha_i g^{ij}\psi^\beta_j.$$

Polynomiality conjecture. If $P$ and $R$ are compatible, meaning that their Schouten bracket vanishes, $[P,R]=0$, then every entry of $Q^{\alpha\beta}$ is a polynomial of degree two in the field variables. This conjecture would provide a tractable condition for compatibility, a problem that the paper says remains out of reach; it is motivated by known bi-Hamiltonian systems of WDVV type.