Painlevé II tau-function expansion from ramified irregular conformal blocks

Let $G$ be the Barnes $G$-function, and let $a$ be a complex number satisfying $a^{3/2}=-\sqrt{-2}$. Let $\Phi^{\Delta,\beta}_{\Lambda',\Lambda}$ denote the indicated ramified irregular vertex operator. Painlevé II expansion conjecture. A series expansion at the irregular singular point $\infty$ is

$$\tau(t)=t^{-\theta^2/2}\sum_{n\in\mathbb{Z}}s^n(2\pi)^{-\nu-n}(4\sqrt{2})^{-(\nu+n)^2}e^{\pi\sqrt{-1}\nu^2/2}a^{-3(\nu+n)^2/2}G\left(1+\nu+n\pm\frac{\theta}{2}\right)\left\langle0\middle|\Phi^{\theta^2/4,4(\nu+n)/3}_{(0,1,0),(0,1,0)}(at^{-1})\middle|(0,1,0)\right\rangle.$$

Namely, $H=(\log\tau(t))'$ satisfies the paper's second Painlevé differential equation. This is presented as a conjectural formula, with no proof or disproof supplied in the source.