Power-law spectrum conjecture for matrix-valued Gaussian multiplicative chaos

Let $M$ be the matrix-valued Gaussian multiplicative chaos measure constructed for $0<\gamma^2<d$, and let $N$ denote the matrix dimension. For $q\in\left(0,\frac{2d}{\gamma^2}\right)$ and $\ell\in(0,1]$, write $C_q>0$ for a constant and define the structure exponent

$$\zeta(q)=\left(d+\frac{\gamma^2}{2}\right)q-\frac{\gamma^2}{2}q^2.$$

Power-law spectrum conjecture. The moments of $M$ satisfy

$$E\left[\operatorname{tr}\left(M(B(0,\ell))^q\right)\right]\simeq C_q\ell^{\zeta(q)}(-\ln\ell)^{\frac{(q-1)(1-N)}{2}}.$$

The previously established integer-moment asymptotics support this formula in their common range, while the displayed expression for general real $q$ is only heuristically derived. If true, it would show that noncommutativity contributes an additional logarithmic factor to the power-law spectrum.