Its–Prokhorov symplectic-form conjecture for isomonodromic systems

Consider a completely integrable Hamiltonian system with Darboux coordinates $\{P_a,Q_a\}$, Hamiltonians $\{H_k\}$ and times $\{t_k\}$. Let $\mathcal{T}$ be the parameter space of times and let $\mathcal{M}$ be the parameter space of monodromy parameters $\{m_{\ell}\}$. Suppose the Darboux coordinates depend on $(t_k,m_{\ell})$. On $\mathcal{T}\times\mathcal{M}$, let $\boldsymbol{\omega}_{0}$ be the $2$-form defined by the source's extended-form construction, and let $\Omega$ be a symplectic structure on $\mathcal{T}\times\mathcal{M}$.

Its–Prokhorov symplectic-form conjecture. There exists a constant $\gamma\in\operatorname{\mathbb{C}}$ such that

$$\boldsymbol{\omega}_{0}=\gamma\Omega.$$

The conjecture proposes that the extended isomonodromic $2$-form is proportional to the symplectic structure naturally associated with the classical Hamiltonian system. Its general validity is not established in the supplied text.