Variants of Kohnen's conjecture for Hermitian modular forms

Let F be a Hermitian cusp form of weight k and of degree 2 over $Q\left(i\right)$ with Fourier-Jacobi coefficients ${\varphi }_m$, $m\in N$. Motivated by a conjecture of W. Kohnen on the growth of Petersson norm of ${\varphi }_m$ in the set-up of Siegel modular forms, we study analogous questions in the set-up of Hermitian modular forms. We first propose a conjecture in this set-up which is analogous to that of Kohnen. We then provide some evidence by proving the conjecture for cusp forms lying in the Hermitian-Maass subspace. We also study certain other related problems.