Symmetry efficiency conjecture

Let $G_{\mathrm{data}} \subseteq G$ act on $M$, and let $\Gamma$ be a continuous boundary whose symmetry group contains $G_{\mathrm{data}}$. For a target precision $\varepsilon>0$, let $m^*(G,\varepsilon)$ denote the minimum fiber dimension for approximating $\Gamma$ by a $G$-equivariant section as in Definition~. Symmetry efficiency conjecture. For every $\varepsilon>0$,

$$m^*(G_{\mathrm{data}},\varepsilon)\leq m^*(G,\varepsilon).$$

This is the remaining main conjecture for the continuous boundary approximation problem: imposing equivariance under the larger structure group should not reduce the fiber dimension required to achieve a given precision. The constraint is essential, since without requiring full $G$-equivariance, inclusion of structure groups would give the opposite inequality trivially.