Dual degenerate insertion moment and beta-factorization conjecture

Let $\gamma\in(0,2)$, let $X(e^{i\theta})$ be the log-correlated Gaussian field appearing in the circular Liouville conformal field theory setting, and let $Y_\gamma$ be the Fyodorov–Bouchaud random variable. Define

$$M_{\gamma}^{\mathrm{dual}}=\frac{1}{2\pi}\int_0^{2\pi}|1-e^{i\theta}|^2e^{\frac{\gamma}{2}X(e^{i\theta})}\,d\theta.$$

Dual degenerate insertion conjecture. For every real $p<\frac{4}{\gamma^2}$,

$$\mathbb{E}[(M_{\gamma}^{\mathrm{dual}})^p]=\frac{\Gamma(1-p\frac{\gamma^2}{4})\Gamma(1+\frac{8}{\gamma^2})\Gamma(1+\frac{4}{\gamma^2}-p)}{\Gamma(1-\frac{\gamma^2}{4})^p\Gamma(1+\frac{4}{\gamma^2})\Gamma(1+\frac{8}{\gamma^2}-p)},$$

and, equivalently,

$$M_{\gamma}^{\mathrm{dual}}\overset{law}{=}Y_\gamma X_2^{-1},$$

where $Y_\gamma$ and $X_2$ are independent and $X_2\sim\mathcal{B}(1+\frac{4}{\gamma^2},\frac{4}{\gamma^2})$. The source presents this as an expected analogous formula for the dual degenerate insertion; it does not provide evidence of resolution.