Conjectural convergent long-distance expansion for the sine-Gordon/Painlevé III3 tau function

Let $r$ be the long-distance variable, let $u$ and $\rho$ be parameters related to monodromy data, and let $G(z)$ denote the Barnes $G$-function. Define

$$\mathcal{G}(\nu,r)=e^{\frac{i\pi\nu^2}{4}}2^{\nu^2}(2\pi)^{-\frac{i\nu}{2}}G(1+i\nu)r^{\frac{\nu^2}{2}+\frac14}e^{\frac{r^2}{16}+\nu r}\mathcal{D}(\nu,r),$$

where

$$\mathcal{D}(\nu,r)\sim 1+\sum_{k=1}^{\infty}D_k(\nu)r^{-k},\qquad r\rightarrow\infty.$$

The parameters $(\nu,\rho)$ are related to the monodromy data, with

$$e^{4\pi i\rho}=\frac{\sin 2\pi\eta}{\sin 2\pi(\sigma+\eta)}.$$

Long-distance expansion conjecture. The sine-Gordon/Painlevé $\mathrm{III}_3$ tau function is given by the convergent series

$$\tau\left(2^{-12}r^4\right)=\chi(\sigma,\nu)\sum_{n\in\mathbb{Z}}e^{4\pi i n\rho}\mathcal{G}(\nu+in,r).$$

This conjecture proposes a convergent completion of the formal long-distance asymptotic expansion and relates the tau function to monodromy data. The supplied text gives no resolution status beyond presenting it as a conjecture.