The maximality conjecture for nm-independence on Polish groups

Let $G$ be a Polish group. An nm-independent relation is the relation \mathrel{\raise0.2ex\text{\ooalign{\hidewidth\vert\hidewidth\cr\raise-0.9ex\text{\smile}}}}^{nm} defined by non-meagerness of the corresponding product of stabilizer sets. A strongly admissible geometry on $G$ is an independence geometry induced by an ideal scheme and satisfying the strong admissibility conditions.

The maximality conjecture. The relation \mathrel{\raise0.2ex\text{\ooalign{\hidewidth\vert\hidewidth\cr\raise-0.9ex\text{\smile}}}}^{nm} is the maximum strongly admissible geometry on $G$.

If true, this would show that nm-independence is intrinsic among strongly admissible geometries for Polish groups. The source gives no resolution of the conjecture.