We solve a conjecture from the first and third authors that claims that fully simple maps, which are maps with non self-intersecting disjoint boundaries, satisfy topological recursion for the exchanged spectral curve \((y, x)\), making use of the topological recursion for ciliated maps (building on a result from Belliard, Eynard, and the second and third authors).

Mathematics Subject Classifications: 05A15, 05A19, 46L54

Keywords: Maps, fully simple maps, enumeration, topological recursion