Jimbo–Miwa–Ueno differential conjecture for hbar-deformed meromorphic connections

Consider an isomonodromic deformation of an $\hbar$-deformed meromorphic connection. Let $\mathbf{t}$ be the deformation times and let $\overline{\omega}$ be the associated Hamiltonian differential. Let $(\mathbf{q},\mathbf{p})$ be Darboux coordinates associated to the apparent singularities and their dual partner on the spectral curve, or Darboux coordinates related to them by $\hbar$-independent transformations. Let $\epsilon$ be a constant, and let $dU(\mathbf{t})$ be a purely time-dependent exact differential.

Jimbo–Miwa–Ueno differential conjecture. One has

$$\omega_{\mathrm{JMU}}(\mathbf{q},\mathbf{p},\mathbf{t})=\epsilon\,\overline{\omega}(\mathbf{q},\mathbf{p},\mathbf{t};\hbar=0)+dU(\mathbf{t}).$$

Here $dU(\mathbf{t})$ is an exact term that could be inserted into the definition of the Hamiltonians. The conjecture generalizes the observation established in the paper's examples and includes the expected freedom from purely time-dependent terms and from the choice of the indicated Darboux coordinates.