Rivin-type realization conjecture for non-contractible angled blocks

Let $\pp$ be a compact, oriented, irreducible, atoroidal $3$-manifold with boundary equipped with an angled block structure, and let $\tilde{\pp}$ denote its universal cover. The angled block structure assigns internal angles $\alpha_i$ and external angles $\varepsilon_i=\pi-\alpha_i$ to dual edges, satisfying $0<\alpha_i<\pi$, with external-angle sum $2\pi$ around each face and sum greater than $2\pi$ around every other simple closed curve in the dual graph that bounds a disk in $\pp$. Rivin-type realization conjecture. The universal cover $\tilde{\pp}$ can be realized as a possibly infinite ideal polyhedron in $\mathbb{H}^3$, with dihedral angles specified by $\pp$, uniquely up to isometry. Rivin's theorem establishes the analogous realization and uniqueness statement for contractible angled blocks, namely angled polyhedra; the conjecture proposes that the result extends to non-contractible angled blocks.