The conjectural boundary two-point function formula in boundary Liouville theory

Let $\gamma$ and $Q$ be the Liouville parameters, with $\beta\in(\frac{\gamma}{2},Q)$ and $\mu,\mu_1,\mu_2>0$. For $i=1,2$, define $\sigma_i$ by

$$\mu_i=\sqrt{\frac{\mu}{\sin(\pi\frac{\gamma^2}{4})}}\cos\left(\pi\gamma\left(\sigma_i-\frac{Q}{2}\right)\right).$$

Let $R(\beta,\sigma_1,\sigma_2)$ denote the boundary two-point function, and let $\Gamma_{\frac{\gamma}{2}}$ and $S_{\frac{\gamma}{2}}$ denote the special functions appearing in the formula. Boundary two-point function conjecture. The probabilistic expression for $R(\beta,\sigma_1,\sigma_2)$ equals

$$\left(\frac{\pi\mu(\frac{\gamma}{2})^{2-\frac{\gamma^2}{2}}\Gamma(\frac{\gamma^2}{4})}{\Gamma(1-\frac{\gamma^2}{4})}\right)^{\frac{Q-\beta}{\gamma}}\frac{\Gamma_{\frac{\gamma}{2}}(\beta-Q)\Gamma_{\frac{\gamma}{2}}(Q-\beta)^{-1}}{S_{\frac{\gamma}{2}}(\frac{\beta}{2}+\sigma_1+\sigma_2-Q)S_{\frac{\gamma}{2}}(\frac{\beta}{2}-\sigma_1-\sigma_2+Q)}$$ $$\times\frac{1}{S_{\frac{\gamma}{2}}(\frac{\beta}{2}+\sigma_2-\sigma_1)S_{\frac{\gamma}{2}}(\frac{\beta}{2}+\sigma_1-\sigma_2)}.$$

This is the second of the two simplest exact formulas predicted for the generalized boundary Liouville theory with bulk and boundary potentials; the supplied passage gives no resolution status.