Thurston's conjecture on horospherical Penrose tilings

Let the $3,5,3 honeycomb be a tessellation of$\operatorname{\mathbb{H}}^3$, and let a **horosphere** be a horospherical surface, identified with a flat Euclidean plane. **Thurston's rough conjecture.** Cutting the$3,5,3 honeycomb along a horosphere and applying a procedure analogous to the cut-and-project method should construct the Penrose tiling on that flat horosphere. The source says that the exact conjecture was apparently not recorded and that the procedure was not made precise, so its precise mathematical formulation and resolution remain open.