Confluent exponentiation conjecture for conformal blocks of type G

Let $b$ be the semiclassical parameter, let $c$ be the central charge, and let $t$, $\theta_0$, $\theta_\bullet$ and $\nu$ be the parameters appearing in the confluent conformal block of type $\mathcal G$. Define the classical block by

$$\widetilde{\mathcal U}\left( t\,\bigl|\,\theta_0,\theta_\bullet,\nu\right)=-\nu t^2+\left(\delta_0+3\nu^2-2\theta_\bullet\nu\right)\ln t+\sum_{n=1}^\infty \widetilde{\mathcal U}_n\left(\theta_0,\theta_\bullet,\nu\right)t^{-2n}.$$

Confluent exponentiation conjecture for type $\mathcal G$. The confluent conformal block exponentiates semiclassically:

$$\mathcal G\left( t\sqrt{\tfrac{2i}{b}}\,\bigl|\,c;\tfrac{i\theta_0}{b},\tfrac{i\theta_\bullet}{b}, \tfrac{i\nu}{b}\right)\stackrel{b\to 0}{\sim}\operatorname{const}\cdot\exp\left\{b^{-2}\widetilde{\mathcal U}\left( t\,\bigl|\,\theta_0,\theta_\bullet,\nu\right)\right\}.$$

This gives the semiclassical form of the type $\mathcal G$ confluent block and connects its large-parameter asymptotics with the classical accessory-parameter problem; the statement remains conjectural in the supplied text.