Selberg distribution conjecture for Bacry–Muzy GMC on the interval

Let $M_\mu(ds)$ denote the Bacry–Muzy GMC measure on the interval, with covariance function $r(t)$ as in the source, and set $\tau=2/\mu$. For parameters $\lambda_1$ and $\lambda_2$, consider the weighted total mass

$$\int_0^1 s^{\lambda_1}(1-s)^{\lambda_2}\,M_\mu(ds).$$

Selberg distribution conjecture. Its distribution has Mellin transform

$$\left(\frac{2\pi\,\tau^{1/\tau}}{\Gamma(1-1/\tau)}\right)^q\frac{\Gamma_2(1-q+\tau(1+\lambda_1)\mid\tau)}{\Gamma_2(1+\tau(1+\lambda_1)\mid\tau)}\frac{\Gamma_2(1-q+\tau(1+\lambda_2)\mid\tau)}{\Gamma_2(1+\tau(1+\lambda_2)\mid\tau)}\frac{\Gamma_2(-q+\tau\mid\tau)}{\Gamma_2(\tau\mid\tau)}\frac{\Gamma_2(2-q+\tau(2+\lambda_1+\lambda_2)\mid\tau)}{\Gamma_2(2-2q+\tau(2+\lambda_1+\lambda_2)\mid\tau)},$$

i.e. it is the Selberg integral probability distribution. This conjecture was previously known in special and general cases according to the source, but it is not resolved in the supplied status information.