Finite-subcomplex relaxation conjecture for ideal polyhedra

Let $\mathcal{D}$ be an infinite cellular decomposition appearing in the Euclidean or hyperbolic existence theorems, with edge angles $\Theta_e$ satisfying conditions (1) and (2), and let condition (3) denote the corresponding face-boundary inequality.

Finite-subcomplex relaxation conjecture. Condition (3) can be relaxed so that it is required to hold only outside a finite sub-complex of $\mathcal{D}$.

This conjecture is motivated by the invariance of graph transience or recurrence under modification of a finite subgraph. Results of He–Schramm and Oh provide supporting evidence, while the stated local conditions remain more readily verifiable than the conditions in Rivin's theorem.