Painlevé I tau-function connection formula across a B-cycle Stokes jump

Let $(\nu^\pm,\rho^\pm)$ be two parameter pairs, and let $\mathscr T^\pm(t,\nu^\pm,\rho^\pm,\hbar)$ be analytic $\tau$-functions of $(P_{\rm I})$ defined on domains containing $t=t_c\pm i\epsilon$, respectively. Let $\operatorname{Li}_2$ denote the dilogarithm, and let the negative real axis be the Stokes direction being crossed. Painlevé I tau-function connection conjecture. When $t$ crosses the negative real axis, the analytic $\tau$-functions satisfy

$$\mathscr T^-(t,\nu^-,\rho^-,\hbar)= \exp\left(\frac{1}{2\pi i}\operatorname{Li}_2\left(e^{2\pi i\rho^+/\hbar}\right)\right) \mathscr T^+(t,\nu^+,\rho^+,\hbar),$$

where

$$(\nu^+,\rho^+)=\left(\nu^- -\frac{\hbar}{2\pi i}\log\left(1-e^{2\pi i\rho^-/\hbar}\right),\rho^-\right).$$

This is a nonlinear connection formula relating the two analytic continuations across the Stokes direction; its rigorous status is not established in the supplied text.