**If we know the combinatorics of a polyhedron, and all but one of its dihedral angles, does that uniquely determine the remaining dihedral angle?**

I’m happy to assume the polyhedron is simply connected, if that helps. I do *not* want to assume that it is convex, because the examples I’m interested in are non-convex.

The only partial result along these lines that I know is that the claim does hold for tetrahedra, by the argument outlined here.

On Mastodon, David Eppstein asked whether there is a fixed bound on the number of edges of a polyhedron that realizes a given Dehn invariant, depending only on the rank of the invariant. I strongly suspect there is no such upper bound.

I have been thinking about a simplified version of this question, where the polyhedra are restricted to polyhedra whose dihedral angles are multiples of $\pi/2$ on all but a single edge. Dmitrii Korshunov's answer below seems to settle this question.

Unfortunately I had somehow deluded myself into thinking that this special case implies the general case, which it does not, or not obviously.