Jerrard–Wetzel–Yuan conjecture on convex polyhedra

Let $\mathcal{P}$ be a convex polyhedron, meaning the convex hull of a finite set of points in $\mathbb{R}^3$ in convex position. A polyhedron is Rupert if there exist $R_1,R_2\in\operatorname{SO}(3)$ and $t\in\mathbb{R}^2$ such that

$$P\circ R_1(\mathcal{P}_C)+t\subset P\circ R_2(\mathcal{P}^{\circ}),$$

where $P:\mathbb{R}^3\to\mathbb{R}^2$ drops the $x$-coordinate and $\mathcal{P}_C$ denotes the set of vertices of the cube. Jerrard–Wetzel–Yuan conjecture. Every convex polyhedron is Rupert. The conjecture was stated hesitantly, and the later construction of the Noperthedron shows that the claim is false: a convex polyhedron exists that is not Rupert.