Immersions in a Quaternionic Grassmannian inducing a given 4-form

Let Gr_k(\H^n) be the Grassmannian manifold of Quaternionic $k$-planes in \H^n and let \gamma^n_k\to Gr_k(\H^n) denote the Stiefel bundle of quaternionic $k$-frames in \H^n. Let $\sigma$ denote the first symplectic Pontrjagin form associated with the universal connection on $\gamma^n_k$. We show that every 4-form $\omega$ on a smooth manifold $M$ can be induced from $\sigma$ by a smooth immersion f:M\to Gr_k(\H^n) (for sufficiently large $k$ and $n$) provided there exists a continuous map f_0:M\to Gr_k(\H^n) which pulls back the cohomology class of $\sigma$ onto that of $\omega$.