The log-Gamma polymer lower-tail large deviation rate function conjecture

Let $Z_n(\theta)$ denote the log-Gamma polymer partition function, let $\psi=\Gamma'/\Gamma$, and for $0<s<-\theta\psi(\theta)$ define $b=b(s,\theta)>0$ as the unique positive solution of

$$\int_{0}^{1} \left(\theta\psi(\theta+iu\theta b/2)+\theta\psi(\theta-i u\theta b/2)+2s\right) \frac{du}{\pi\sqrt{1-u^2}}=0.$$

Define

$$f(s,\theta)=b^2 \int_{0}^{1}\left(\theta\psi(\theta+iu\theta b/2)+\theta\psi(\theta-iu\theta b/2)+2s\right)\sqrt{1-u^2}\frac{du}{2\pi},$$

and

$$F(s,\theta)=-\int_{s}^{-\theta\psi(\theta)}f(t;\theta)\,dt.$$

The log-Gamma polymer large deviation conjecture. There exists $\theta_0>0$ such that uniformly for $0<\theta<\theta_0$ and $\epsilon<s<-\theta\psi(\theta)$ for any $\epsilon>0$, we have

$$\lim_{n\to\infty}\frac{-1}{n^2}\log\mathbb P\left[\log Z_n(\theta)\leq \frac{2n}{\theta}s\right]=F(s,\theta).$$

This conjectures the lower-tail large deviation rate function for the log-Gamma polymer in the small-$\theta$ regime. The supplied text gives no resolution, so the conjecture remains open.