Given a positive Laplace eigenfunction on a hyperbolic manifold, there exists a horospherically invariant measure corresponding to it, known as the Burger-Roblin measure. A question attributed to Babillot concerns whether every horospherically invariant measure arises from such a correspondence. In this talk, I will survey previous works where affirmative answers to this question were found and present a new result extending it to a broad class of subgroups in rank one Lie groups. This is joint work with Or Landesberg, Elon Lindenstrauss, and Hee Oh.