Regular homotopy of Hurwitz curves

We prove that any two irreducible cuspidal Hurwitz curves $C_0$ and $C_1$ (or, more generally, two curves with $A$-type singularities) in the Hirzebruch surface $\boldsymbol F_N$ with the same homology classes and sets of singularities are regular homotopic. Moreover, they are symplectically regular homotopic if $C_0$ and $C_1$ are symplectic with respect to a compatible symplectic form.