Knizhnik-Zamolodchikov equations in Deligne categories

We consider the Knizhnik-Zamolodchikov equations in Deligne Categories in the context of $(\mathfrak{gl}_m,\mathfrak{gl}_{n})$ and $(\mathfrak{so}_m,\mathfrak{so}_{2n})$ dualities.

We derive integral formulas for the solutions in the first case and compute monodromy in both cases.