Its–Prokhorov rational primitive conjecture for the extended isomonodromic form

Consider a completely integrable Hamiltonian system with Darboux coordinates $\{P_a,Q_a\}$, Hamiltonians $\{H_k\}$ and times $\{t_k\}$. Let $\boldsymbol{\omega}_{cla}$ be the classical action differential on the time and monodromy parameter space, and let $\boldsymbol{\omega}_{0}$ be the extended $2$-form used in the preceding conjecture. Let $\gamma$ be the constant appearing in the proportionality relation between $\boldsymbol{\omega}_{0}$ and the symplectic structure.

Its–Prokhorov rational primitive conjecture. There exists a function $G(P_a,Q_a,t_k)$, rational in the variables $\{P_a\}$, $\{Q_a\}$ and $\{t_k\}$, such that

$$\boldsymbol{\omega}_{0}=\gamma\boldsymbol{\omega}_{cla}+dG.$$

The assertion says that the extended form differs from a scalar multiple of the classical action differential by an exact differential with a rational primitive. The supplied text does not state a resolution of this conjecture.