Stokes automorphism conjecture for the Painlevé I perturbative partition function

Let $Z(t,\nu,\hbar)$ be the perturbative partition function for the Painlevé I system, and let $\mathfrak S$ denote the Stokes automorphism associated with the $B$-periods. Write $\partial_\nu$ for differentiation with respect to $\nu$, and interpret functions of $e^{-\hbar\partial_\nu}$ as formal difference operators. Stokes automorphism conjecture. The Stokes automorphism acts on $Z$ by

$$\mathfrak S Z(t,\nu,\hbar)= \exp\left( \frac{1}{2\pi i}\operatorname{Li}_2\left(e^{-\hbar\partial_\nu}\right) -\frac{\hbar\partial_\nu}{2\pi i}\log\left(1-e^{-\hbar\partial_\nu}\right) \right)Z(t,\nu,\hbar).$$

This is the zero-Fourier-mode consequence of the preceding conjectural tau-function connection formula and describes the resurgent Stokes jump of the perturbative partition function; its status is not resolved in the supplied text.