Subgroup projection-distance conjecture for closed orthogonal subgroups

Let $\mathcal{G}$ be a closed subgroup of the orthogonal group $\mathcal{O}(d)$, and let $\operatorname{conv}(\mathcal{G})$ denote its convex hull. For $X\in\operatorname{conv}(\mathcal{G})$, let $\operatorname{dist}(X,\mathcal{G})$ be the distance from $X$ to $\mathcal{G}$, and let $\Pi_{\mathcal{G}}(X)$ be the projection of $X$ onto $\mathcal{G}$. Subgroup projection-distance conjecture. There exists a constant $\rho\geq 1$ such that

$$\operatorname{dist}(X,\mathcal{G})\leq \rho\operatorname{Tr}\left(I_d-X^\top\Pi_{\mathcal{G}}(X)\right)\quad\forall X\in\operatorname{conv}(\mathcal{G}).$$

The conjecture would extend the geometric inequality established in the paper for the orthogonal, special orthogonal, permutation, and cyclic groups to every closed subgroup of the orthogonal group. Its resolution would provide the condition needed for the paper's master theorem to apply uniformly across arbitrary closed orthogonal subgroups.