Morris distribution conjecture for Bacry–Muzy GMC on the circle

Let $M_\mu(ds)$ be the Bacry–Muzy GMC measure on the circle, let $\tau=2/\mu$, and let $M_{(\tau,\lambda,\lambda)}$ denote the corresponding special case of the Morris integral probability distribution. Consider the weighted total mass

$$\int_0^1 \lvert 1-e^{2\pi i s}\rvert^{2\lambda}\,M_\mu(ds).$$

Morris distribution conjecture.

$$\int_0^1 \lvert 1-e^{2\pi i s}\rvert^{2\lambda}\,M_\mu(ds)=M_{(\tau,\lambda,\lambda)}.$$

The Morris distribution has the required moment and Mellin-transform properties in the cases discussed in the paper, but the identification with the GMC total mass remains conjectural in the supplied source.