I am currently writing a geometry paper "Rectifications of Convex Polyhedra" and I am confused to have discovered what appears to be a remarkable discrete geometric fact:

**Conjecture: Let $P$ be a convex polyhedron. Then $P \cong P^{\circ} \sqcup R_{1}[P]$.**

The relation $\cong$ is to denote scissors congruence, the dual polyhedron is $P^{\circ}$, and $R_{1}[P]$ is the first rectification of $P$, defined as $$R_{1}[P] = \text{conv} \left\{\frac{x_{i} + x_{j}}{2} \; | \; (i,j) \in E(P) \right\}.$$

I have been drawing many examples of when this happens to work out (cube decomposing into a rectified cube (cuboctahedron) and eight pieces (orthoschemes) which rearrange to be an octahedron; octahedron decomposing into a rectified octahedron (cuboctahedron) and six pieces which rearrange to be a cube; tetrahedron decomposing into a rectified tetrahedron (octahedron) and four pieces which rearrange to be a tetrahedron. I've also tried various examples of prisms and bipyramids which all appear to work, I believe if a counterexample exists it will be non-centrally symmetric and I am working on a proof in the case of centrally symmetric convex polyhedra.

I am interested in classifying which convex polyhedra this holds for if there is a counterexample to the conjecture.