Correlations of $C$ and $CP$ violation in $\eta \to {\pi }^0{\mathrm{\ell }}^{+}{\mathrm{\ell }}^{-}$ and ${\eta }^{\mathrm{\prime }}\to \eta {\mathrm{\ell }}^{+}{\mathrm{\ell }}^{-}$

Based on recent progress in the systematic analysis of $C$ and $CP$ violation in the light-meson sector, we calculate the $C$-odd transition amplitudes $\eta \to {\pi }^0{\mathrm{\ell }}^{+}{\mathrm{\ell }}^{-}$ and ${\eta }^{\mathrm{\prime }}\to \eta {\mathrm{\ell }}^{+}{\mathrm{\ell }}^{-}$. Focusing on long-distance contributions driven by the lowest-lying hadronic intermediate states, we work out the correlations between these beyond-the-Standard-Model signals and the Dalitz-plot asymmetries in $\eta \to {\pi }^0{\pi }^{+}{\pi }^{-}$ and ${\eta }^{\mathrm{\prime }}\to \eta {\pi }^{+}{\pi }^{-}$, using dispersion theory.