In this talk we briefly reviewed our recent progress on a dispersive study of $\pi\pi \rightarrow K \bar K$ scattering data. First, we optimized the applicability of hyperbolic dispersion relations
to this process and we have then showed the existence of
sizable inconsistencies on the existing data sets.
However, by imposing these dispersion relations as constraints on data fits, we are able to provide a simple set of parameterizations for the S, P and D partial waves up to 2 GeV that describe the data while being consistent with dispersion relations up to their maximum applicability limit of 1.47 GeV.