Injective maps between flip graphs

We prove that every injective simplicial map $$\mathcal{F}$(S) \to $\mathcal{F}$(S’)$betweenflipgraphsisinducedbyasubsurfaceinclusion$S\to S’$, except in finitely many cases. This extends a result of Korkmaz–Papadopoulos which asserts that every automorphism of the flip graph of a surface without boundary is induced by a surface homeomorphism.