Uniform hyperbolicity of the graphs of nonseparating curves via bicorn curves

We show that the graphs of nonseparating curves for oriented finite type surfaces are uniformly hyperbolic. Our proof closely follows the proof of uniform hyperbolicity of the graphs of curves for closed surfaces due to Przytycki-Sisto. As demonstrated by Aramayona-Valdez, this proves also that the graph of nonseparating curves for any oriented infinite type surface with finite genus $\geq 2$ is hyperbolic.