An Open Problem — or Two ...

Suppose $E$ is a Hilbert $B$-module and $\Phi$ is a bounded right-linear map from $E$ to $B$. Suppose, further, that $F$ is a zero-complemented submodule of $E$. Does $\Phi(F)=\{0\}$ imply $\Phi(E)=\{0\}$?
To our knowledge, this natural question is an open problem. We recollect our thoughts about it. These include the formulation of some statements that, when true, would prove the affirmative answer, or that, when false, would disprove it.