The condition ii) in the theorem can be replaced by the following small non-vanishing $n$-cycle condition: there is an $[A]\in H_{n}(B_{g}(p,\sigma i_{g}(p)),\mathbb{Z})$ such that

$$\int_{A}\Omega\neq 0.$$

This condition can not be removed since it is satisfied if there is a special lagrangian submanifold $L$ near $p$ having comparable size to $i_{g}(p)$, for example $L\subset B_{g}(p,\sigma i_{g}(p))$.