Almost-polynomial moving-frame conjecture for orthogonal invariants

For $d\geq 2$, let $\mathcal U_{d;\leq n}$ be the domain of the moving frame and let

$$\rho_d:\mathcal U_{d;\leq n}\to\operatorname{O}_d(\mathbb{R})$$

be the moving frame defined in the preceding section. Almost-polynomial moving-frame conjecture. For every $d\geq 2$, the moving frame $\rho_d$ is almost-polynomial. The paper states that this conjecture holds for $d=3$, while it remains open for $d>3$.