The Axiom of Choice, and the statement that whenever there is a surjection $A \to B$ there is an injection $B \to A$

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Is there a model of $\mathsf{ZF}$ in which the Axiom of Choice fails but the following statement holds for arbitrary sets $A, B$:

there is an injection $A \to B$ if there is a surjection $B \to A$?