The symmetry-group decomposition conjecture for convex bodies

Let $K\subset\mathbb R^n$, $n\geq 3$, be a convex body, and let $G<\operatorname{O}(\mathbb R^n)$ be its symmetry group. Let $\pi_k$ and $\rho_k$ denote the subgroup types defined in the source. Symmetry-group decomposition conjecture. The group $G$ can be written as a set-theoretic union

$$G=A_1\cup A_2\cup\cdots\cup A_s\cup B_1\cup B_2\cup\cdots\cup B_t,$$

where $A_i\leq G$ is a subgroup conjugated to $\pi_{k_i}$ for $i=1,2,\ldots,s$, and $B_i\leq G$ is a subgroup conjugated to $\rho_{k_i}$ for $i=1,2,\ldots,t$. The conjecture proposes a restricted structure for symmetry groups of convex bodies, but the source gives no resolution.