Measured foliations at infinity of quasi-Fuchsian manifolds

Let $({\lambda }^{+}(M),{\lambda }^{-}(M))$ denote the pair of measured foliations at the boundary at infinity ${\mathrm{\partial }}_{\mathrm{\infty }}$ of a quasi-Fuchsian manifold $M$. We prove that $({\lambda }^{+}(M),{\lambda }^{-}(M))$ is filling if $M$ is close to being Fuchsian. We also show that given any filling pair $({\alpha }_1,{\alpha }_2)$ of measured foliations, and every small enough $t>0$, the pair $(t{\alpha }_1,t{\alpha }_2)$ is realised as the pair of measured foliations at infinity of some quasi-Fuchsian manifold $M$. This answers questions of Schlenker near the Fuchsian locus.