The inner phase II complex Gaussian multiplicative chaos conjecture

Let $M^{\gamma,\beta}_\varepsilon$ be the complex Gaussian multiplicative chaos measure and let $M'$ be the derivative martingale. For $\beta>0$ and $\gamma>\sqrt{\frac d2}$ satisfying $\gamma+\beta>\sqrt{2d}$, set $\alpha=\sqrt{\frac d2}\frac1\gamma$. Let $N^\alpha_{M'}$ be an independently scattered $\alpha$-stable random measure with intensity $M'$, characterized by

$$\mathds{E}[e^{-qN^\alpha_{M'}(A)}]=e^{-q^\alpha M'(A)}$$

for all $q\;\geqslant\;0$ and bounded Borel sets $A$, and, conditionally on $N^\alpha_{M'}$, let $W_{\sigma^2N^\alpha_{M'}}$ be a complex Gaussian random measure with intensity $N^\alpha_{M'}$.

Inner phase II conjecture. The following convergence in law holds:

$$\left((\ln\frac1\varepsilon)^{\frac{3\gamma}{2\sqrt{2d}}}\varepsilon^{\gamma\sqrt{2d}-d}M^{\gamma,\beta}_\varepsilon(A)\right)_{A\subset\mathbb{R}^d} \Rightarrow \left(W_{\sigma^2N^\alpha_{M'}}(A)\right)_{A\subset\mathbb{R}^d},$$

where $\sigma^2$ depends on $(\gamma,\beta)$.

This conjecture describes the limiting complex Gaussian random measure in the inner phase II of complex Gaussian multiplicative chaos. The source notes analogous evidence from complex branching Brownian motion, but does not report a proof for Gaussian multiplicative chaos.