Voros connection formula conjecture for the Painlevé I WKB solution

Fix $(t,\nu)\in D_{t_\ast}\times D_{\nu_\ast}$ so that the Stokes graph has no Stokes segment. Let $C$ be a Stokes curve connecting a branch point $e$ and $\infty$, forming a common boundary of Stokes regions $I$ and $II$, with $II$ next to $I$ counterclockwise around $e$. Let $\psi_{\pm}^{J}$ be the Borel sum of $\psi_{\pm}$ on region $J$, and let $\widetilde{\psi}_{\mp}^{II}$ be the Borel sum of the termwise analytically continued detoured WKB series described in the source. Voros connection formula conjecture. If

$$\operatorname{Re}\int_e^x\sqrt{4x^3+2tx+u(t,\nu)}\,dx>0$$

on $C$, then

$$\psi_+^I=\psi_+^{II}+\widetilde{\psi}_-^{II},\qquad \psi_-^I=\psi_-^{II}.$$

If instead

$$\operatorname{Re}\int_e^x\sqrt{4x^3+2tx+u(t,\nu)}\,dx<0$$

on $C$, then

$$\psi_+^I=\psi_+^{II},\qquad \psi_-^I=\psi_-^{II}+\widetilde{\psi}_+^{II}.$$

The formula is expected to hold for the WKB solution of the BPZ equation, extending the usual Voros connection formula from Schrödinger-type equations; its validity is not proved in the supplied text.