  Ordinary maps satisfy topological recursion for a certain spectral curve $(x,
y)$. We solve a conjecture from arXiv:1710.07851 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 arXiv:2105.08035.
