Testing epimorphism onto the bicyclic monoid is $\mathsf{N}\mathsf{P}$-complete

We prove that deciding whether there is a surjective homomorphism from an arbitrary finitely presented inverse monoid onto the bicyclic monoid is $\mathsf{N}\mathsf{P}$-complete. As part of the proof, we show that an extension of existential Presburger arithmetic which involves greatest common divisors on $n$ arguments is in $\mathsf{N}\mathsf{P}$, extending a recent result of Défossez, Haase, Mansutti, and Pérez (SODA 2024).