Steininger–Yurkevich conjecture on the rhombicosidodecahedron

Let $\mathcal{P}$ be a polyhedron. It is Rupert if there exist $R_1,R_2\in\operatorname{SO}(3)$ and $t\in\mathbb{R}^2$ such that

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

where $P:\mathbb{R}^3\to\mathbb{R}^2$ drops the $x$-coordinate and $\mathcal{P}^{\circ}$ is the interior of $\mathcal{P}$. Steininger–Yurkevich conjecture. The rhombicosidodecahedron is not Rupert. This remains open to the best of the authors' knowledge, following the Noperthedron counterexample to the conjecture that all convex polyhedra are Rupert.