The discriminant-form surjectivity conjecture for invariant lattices

Let $L_G$ be a lattice of type (a) or (b), let $L^G$ be the associated invariant lattice, let $A_{L^G}$ be its discriminant group, and let $\overline{O}(L^G)$ denote the image of its orthogonal group in $O(A_{L^G})$. Discriminant-form surjectivity conjecture. For $L_G$ a lattice of type (a) or (b) and $\operatorname{rk} L^G\geq 4$, we have

$$\overline{O}(L^G)=O(A_{L^G}).$$

The claim concerns the lattice-theoretic classification of invariant lattices associated with finite symplectic automorphism groups; the surrounding discussion indicates that the result is established by explicit uniqueness arguments, but the supplied text does not explicitly state its resolution status.