The interaction of light's orbital angular momentum (OAM) with matter has still several unexplored aspects. In particular, it is unknown if there exists for OAM an effect analogous to spin angular momentum-based optical activity. Here we study experimentally the influence of OAM on the interaction of light with a cholesteric liquid crystal polymer. We use strongly focussed light where the polarization and the orbital degrees of freedom are coupled. Two possible manifestations of an OAM-sensitive interaction are investigated: (i) the modification of circular dichroism, and (ii) the occurrence of intermodal dispersion of the { = +1, = −1} modes. We conclude that such an interaction does not exist within the experimental parameter range studied here.