Uniqueness of generalized conformal restriction measures and Malliavin-Kontsevich-Suhov measures for $c\in (0,1]$

In this paper, we present a unified approach to establish the uniqueness of generalized conformal restriction measures with central charge $c\in (0,1]$ in both chordal and radial cases, by relating these measures to the Brownian loop soup. Our method also applies to the uniqueness of the Malliavin-Kontsevich-Suhov loop measures for $c\in (0,1]$, which was recently obtained in [Baverez-Jego, arXiv:2407.09080] for all $c\le 1$ from a CFT framework of SLE loop measures. In contrast, though only valid for $c\in (0,1]$, our approach provides additional probabilistic insights, as it directly links natural quantities of MKS measures to loop-soup observables.