Given a locally compact group G, Leptin's theorem states that G is amenable if and only if the Fourier algebra A(G) admits a bounded approximate identity, where the latter property is known as co-amenability of the quantum dual of G. In the quantum setting, this characterization is known as the duality between amenability and co-amenability. It is proved that a discrete quantum group is amenable if and only if its dual compact quantum group is co-amenable. The definition of co-amenability for quantum homogeneous spaces is given by Kalantar-Kasprzak-Skalski-Vergnioux. Furthermore, they ask whether the co-amenability of a quantum homogeneous space is equivalent to the (relative) amenability of its co-dual. In this talk, we will answer this question for quantum homogeneous spaces of compact Kac quantum groups under a mild assumption. Based on joint work with Mehrdad Kalantar.