Bijectivity conjecture for orthogonal maps fixing the radical

Let $R$ be a commutative ring and let $(M,q)$ be a quadratic module over $R$. Write $M^{\perp}$ for the radical of the polar form of $q$, and let an orthogonal map mean an $R$-linear map preserving $q$. Bijectivity conjecture. Every orthogonal map from $M$ to itself that restricts to $\operatorname{Id}_{M^{\perp}}$ on $M^{\perp}$ is bijective, hence lies in $\operatorname{O}_{M^{\perp}}(M,q)$. This is motivated by cases where injectivity implies surjectivity, including finite-dimensional quadratic spaces over fields and suitable free modules; the claim is posed for arbitrary commutative rings and quadratic modules, with no resolution given here.