Let $V$ be a symplectic space over a field $k$ (for simplicity, the characteristic of $k$ is not $2$). The Maslov index sends a collection of $n$ lagrangian subspaces of $V$ to a quadratic space over $k$ (see Teruji Thomas's excellent paper on this).

My question: is this index surjective on (isometry classes of nondegenerate) quadratic spaces as we allow $n$ and the choice of lagrangians to vary, or if not then which quadratic spaces it hits? I don't think it matters, but I'm interested in the case where $k$ is a non-Archimedean local field.