Superrigidity for representations of transverse measured groupoids

For $i=1,\dots ,k$, let ${\mathbf{G}}_i$ be a connected, simply connected, semisimple algebraic group over some local field ${\kappa }_i$ of characteristic zero. Let $G_i={\mathbf{G}}_i({\kappa }_i)$ be the ${\kappa }_i$-points of ${\mathbf{G}}_i$ and denote by $G={\prod }_{i=1}^k{G}_i$. If we assume that $G$ has higher rank and each factor has positive rank, given an ergodic transverse $G$-system $(X,\mu ,Y)$, we prove a superrigidity phenomenon for Zariski dense representations of the transverse groupoid $(G⋉X){\mathrm{\mid }}_Y$ into either an almost simple or a reductive algebraic group.