How does the supercritical GMC converge?

In the spirit of [M. Biskup & O. Louidor, Adv. Math. 330 (2018)], we study the local structure of $\star$-scale invariant fields -- a class of log-correlated Gaussian fields -- around their extremal points by characterising the law of the "shape" of the field's configuration near such points. As a consequence, we obtain a refined understanding of the freezing phenomenon in supercritical Gaussian multiplicative chaos.