Generalized Andreev conjecture for trivalent hyperbolic polyhedra

Let $P$ be an abstract trivalent polyhedron with more than five faces, and let $\Theta:\mathcal{E}\to(0,\pi)$ be a dihedral-angle function. A Whitehead pair is a pair of edges of $P$ belonging to a common face and having four distinct endpoints. A prismatic $k$-circuit is a curve in the dual complex intersecting $k$ edges of $P$ and forming the usual prismatic circuit. Assume the following conditions hold:

• If $\Gamma$ is the boundary of the union of two adjacent triangles of $P^*$ and intersects edges $e_1,e_2,e_3,e_4$, with $e_1,e_2$ and $e_3,e_4$ both Whitehead pairs, then either $\Theta(e_1)+\Theta(e_2)\leq\pi$ or $\Theta(e_3)+\Theta(e_4)\leq\pi$.
• Whenever distinct edges $e_1,e_2,e_3$ meet at a vertex,

$$\sum_{\mu=1}^3\Theta(e_\mu)>\pi,$$

and

$$\Theta(e_1)+\Theta(e_2)<\Theta(e_3)+\pi,\qquad \Theta(e_2)+\Theta(e_3)<\Theta(e_1)+\pi,\qquad \Theta(e_3)+\Theta(e_1)<\Theta(e_2)+\pi.$$

• For every prismatic $k$-circuit $\Gamma$ intersecting edges $e_1,\ldots,e_k$,

$$\sum_{\mu=1}^k\Theta(e_\mu)<(k-2)\pi.$$

Generalized Andreev conjecture. There exists a compact convex hyperbolic polyhedron $Q$ combinatorially equivalent to $P$ whose dihedral angles are given by $\Theta$, and $Q$ is unique up to isometries of $\mathbb H^3$. The conjecture would extend Andreev's characterization of compact convex hyperbolic polyhedra to this class of trivalent polyhedra, with the Whitehead-pair condition addressing the additional combinatorial obstruction. The supplied text does not state whether the conjecture has been resolved.