Bohr–Sommerfeld conjecture for the type G classical confluent conformal block

Let $\nu$ be the Bohr–Sommerfeld period along the specified cycle, and let $\mathcal E^{[\mathrm{BS}]}(t\,|\,\theta_0,\theta_\bullet,\nu)$ denote the corresponding accessory parameter. Let $\widetilde{\mathcal U}(t\,|\,\theta_0,\theta_\bullet,\nu)$ be the classical confluent conformal block of type $\mathcal G$.

Bohr–Sommerfeld conjecture for type $\mathcal G$.

$$\mathcal E^{[\mathrm{BS}]}\left( t\,|\,\theta_0,\theta_\bullet,\nu\right)=\frac{\partial}{\partial t}\widetilde{\mathcal U}\left( t\,\bigl|\,\theta_0,\theta_\bullet,\nu\right).$$

This relates the Bohr–Sommerfeld accessory parameter for the fourth Painlevé-type confluent Heun problem to the derivative of the classical type $\mathcal G$ block; the supplied text reports perturbative comparison but no resolution of the conjecture.