In their preprint about the Shafarevich conjecture for hypersurfaces on abelian varieties, Lawrence and Sawin prove a big monodromy theorem for families of hypersurfaces by reducing it to a similar result for Tannaka groups of perverse intersection complexes. A large part of their work is an intricate combinatorial argument about Hodge numbers, which is used to exclude that the Tannaka group acts via wedge powers of the standard representation of SL(n). We explain a simple geometric proof of the analogous result when hypersurfaces are replaced by subvarieties of high codimension; this is joint work in progress with Ariyan Javanpeykar, Christian Lehn and Marco Maculan.