Self-duality conjecture for the Bacry–Muzy GMC Mellin transform

Let $M_\mu$ be the Bacry–Muzy GMC measure on the interval and set $\tau=2/\mu$. Define its total-mass Mellin transform by

$$\mathfrak{M}(q\mid\tau)\triangleq {\bf E}\left[\left(\int_0^1 M_\mu(ds)\right)^q\right].$$

Self-duality conjecture. Under $\tau\mapsto1/\tau$ and $q\mapsto q/\tau$, it satisfies

$$\mathfrak{M}\left(\frac{q}{\tau}\mid\frac{1}{\tau}\right)(2\pi)^{-q/\tau}\Gamma^{q/\tau}(1-\tau)\Gamma\left(1-\frac{q}{\tau}\right)=\mathfrak{M}(q\mid\tau)(2\pi)^{-q}\Gamma^q\left(1-\frac{1}{\tau}\right)\Gamma(1-q).$$

The source notes that the identity is known for the Morris and Selberg integral probability distributions, while presenting it here as a GMC conjecture.