Cocycle superrigidity for profinite actions of irreducible lattices

Let $\Gamma$ be an irreducible lattice in a product of two locally compact groups and assume that is densely embedded profinite group $K$. We give necessary conditions which imply the left translation action $\Gamma \curvearrowright K$ “virtually” cocycle superrigid: any ${w\colon \Gamma\times K\rightarrow\Delta}$ with values countable $\Delta$ cohomologous to factors through map $\Gamma\times K\rightarrow\Gamma\times K\_0$ for some finite quotient $K\_0$ As corollary, we deduce ergodic $\Gamma=\mathrm{SL}\_2(\mathbb Z\[S^{-1}])$ virtually superrigid W$^\*$-superrigid nonempty set primes $S$.