The triple-point complex Gaussian multiplicative chaos conjecture

Let $M^{\gamma,\beta}_\varepsilon$ be the complex Gaussian multiplicative chaos measure and let $M'$ be the derivative martingale. At the triple point, take

$$\beta=\gamma=\sqrt{\frac d2}.$$

Conditionally on $M'$, let $W_{\sigma^2M'}$ be a complex Gaussian random measure with intensity $\sigma^2M'$, where $\sigma^2$ is a constant.

Triple-point conjecture. As $\varepsilon\to0$,

$$\left(|\ln\varepsilon|^{-\frac14}M^{\gamma,\beta}_\varepsilon(A)\right)_{A\subset\mathbb{R}^d}\stackrel{\varepsilon\to0}{\Rightarrow}\left(W_{\sigma^2M'}(A)\right)_{A\subset\mathbb{R}^d}.$$

The conjecture concerns the delicate triple-point regime, where all intermediate scales contribute to the limiting second moment. The source explains that weak convergence of the relevant approximations makes simultaneous control of all scales difficult and gives no resolution.