We construct certain orbifold compactifications of the moduli stack of pointed stable curves over $\mathbb C$ and study their fundamental groups by means of their quantum representations. This enables to construct interesting K\"ahler groups and to settle most of the candidates for a counter-example to the Shafarevich conjecture on holomorphic convexity proposed in 1998 by Bogomolov and Katzarkov, using TQFT representations of the mapping class groups.