Let $A$ be a $C^*$-algebra and $E$ be a (right) Hilbert $C^*$-module over $A$. Assume $F$ is a closed submodule of $E$ such that $F^\perp := \{x \in E: \langle x, F\rangle=0\}$ is orthogonally complemented, i.e. we have $F^\perp \oplus F^{\perp \perp} = E.$ Can we conclude that $F$ is orthogonally complemented, and if this is the case do we have $F= F^{\perp \perp}$?

I know that in general we don't need to have $F= F^{\perp \perp}$. However, when $F$ is orthogonally complemented, i.e. $F\oplus F^\perp = E$ this is obvious. For Hilbert spaces (i.e. $A = \mathbb{C}$), there are no counterexamples.