We prove that the G\"ottsche-Schroeter and Schroeter-Shustin refined invariants specialize at $q=1$ to the enumeration of rational, resp. elliptic complex curves on arbitrary toric surfaces matching constraints that consist of points and of points with a contact element. Furthermore, we show that the refined invariant extends to the case of any genus g≥2 and either one contact constraint or points in Mikhalkin position, and it again specializes to the corresponding characteristic number at $q=1$. In the appendix we show the limitations of extending this count to a more general setting.
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Shustin et al. (2024) studied this question.