Motivated by the recent appearance of the trillium lattice in the search for materials hosting spin liquids, we study the ground state of the classical Heisenberg model on its line graph, the trilline lattice. We find that this network realizes the recently proposed notion of a fragile spin liquid in three dimensions. Additionally, we analyze the Ising case and argue for a possible ℤ2 quantum spin liquid phase in the corresponding quantum dimer model. Like the well-known 𝑈(1) spin liquids, the classical phase hosts moment fractionalization evidenced by the diluted lattice, but unlike those, it exhibits exponential decay in both spin correlations and interactions between fractionalized moments. This provides an instance of a purely short-range correlated classical Heisenberg spin liquid in three dimensions.
