The Painlevé IV conformal-block expansion conjecture

Let $\theta$, $\theta_t$, $\beta$, and $s$ be parameters, let $G$ denote the Barnes $G$-function, and let $\Phi$ and the matrix elements appearing below be the conformal-block data. Define

$$\tau(t)=t^{-2\theta_t^2}e^{\theta_t t^2}\sum_{n\in\mathbb{Z}} s^n G(1+\theta-\beta-n) \prod_{\epsilon=\pm 1}G(1+\theta_t+\epsilon(\beta+n))\left\langle 0\middle|\cdot\left(\Phi^{\theta_t^2}_{(\theta,0,1/4),(\theta-\beta-n,0,1/4)}(1/\sqrt{2}t)\middle|(\theta,0,1/4)\right)\right\rangle.$$

Set $H=(\log\tau(t))'$. Painlevé IV conformal-block conjecture. The function $H$ satisfies the differential equation

$$\left(H”\right)^2-4(tH'-H)^2+4H'(H'-2(\theta+\theta_t))(H'-4\theta_t)=0.$$

This proposes a conformal-block expansion for a tau-function of the Painlevé IV Hamiltonian equation. The source presents the formula as a conjecture, but supplies no resolution status beyond its assertion.