Thurston's modified conjecture on horospherical sections of the {3,5,3} honeycomb

Take the $\{3,5,3\}$ honeycomb in $\operatorname{\mathbb{H}}^3$ and cut it along a horosphere. Consider the union of all icosahedra interior to, or intersecting, the horosphere; this defines a patch $S_i$ of the $\{3,5,3\}$ honeycomb in $\operatorname{\mathbb{H}}^3$ with infinite boundary $\partial S_i$. Thurston's modified conjecture. This boundary $\partial S_i$ is equivalent to, or seeds the growth of, a Penrose tiling. The exact meaning of equivalence and the proposed construction are not made precise, and it is unclear whether this is merely a rephrasing of Thurston's rough conjecture or an essential change.