The weak Stoker conjecture for polyhedra

Let $P_1$ and $P_2$ be polyhedra with the same $1$-skeleton $\Gamma$. Suppose that, for every edge $e$ of $\Gamma$, the dihedral angles of $P_1$ and $P_2$ at the corresponding edge are equal. Weak Stoker conjecture. Then $P_1$ and $P_2$ have the same volume. The paper introduces this as a potentially more tractable formulation and proves that it is equivalent to the Stoker conjecture, so it remains open together with the latter.