Fourier-transform asymptotic expansion conjecture for the Painlevé I tau function

Let $q(t\,|\,\boldsymbol\nu)$ be a Painlevé I function with Stokes parameters $\boldsymbol\nu$, and let $\mathcal R_k$ be a ray on which $t\to\infty$. Define

$$x=24^{\frac14}\left(e^{\frac{2\pi i k}{5}-\pi i}t\right)^{\frac54}.$$

Let $G(z)$ denote the Barnes $G$-function, and let $\mathcal C_k(\boldsymbol\nu)$, $B_j(\nu)$ be the coefficients appearing in the asymptotic expansion. Fourier-transform asymptotic expansion conjecture. The asymptotic expansion of the Painlevé I tau function $\tau(t\,|\,\boldsymbol\nu)$ as $t\to\infty$ along $\mathcal R_k$ has the structure

$$\tau(t\,|\,\boldsymbol\nu)\simeq \mathcal C_k(\boldsymbol\nu)x^{-\frac{1}{60}}e^{\frac{x^2}{45}}\sum_{n\in\mathbb Z}e^{2\pi in\nu_{k+1}}\mathcal B(\nu_k+n,x),$$

where

$$\mathcal B(\nu,x)\simeq C(\nu)x^{-\frac{\nu^2}{2}}e^{\frac45 i\nu x}\left[1+\sum_{k=1}^{\infty}\frac{B_k(\nu)}{x^k}\right],$$

and

$$C(\nu)=48^{-\frac{\nu^2}{2}}(2\pi)^{-\frac\nu2}e^{-\frac{i\pi\nu^2}{4}}G(1+\nu).$$

The formula is a precise conjectural description of the Painlevé I tau-function asymptotics, refining the corresponding asymptotics of the Painlevé I solution. It was verified by explicitly calculating more than 50 terms in the asymptotic expansion of $\tau(t\,|\,\boldsymbol\nu)$, so the conjecture is considered solved in the source.