Separation conjecture for Hilbert C*-submodules by bounded modular functionals

Let $A$ be a C*-algebra, and let $\mathcal M$ be a Hilbert $A$-submodule of a Hilbert $A$-module $\mathcal N$. Suppose that the orthogonal complement of $\mathcal M$ relative to $\mathcal N$ is trivial:

$$\mathcal M^\perp=\{0\}.$$

Separation conjecture. There does not exist any non-trivial bounded $A$-linear map from $\mathcal N$ to $A$ that is equal to the zero map on $\mathcal M$.

This is equivalent to asking whether adequate bounded modular functionals always separate Hilbert C*-submodules, or equivalently whether non-trivial extensions of the zero modular functional exist. It is also related to the existence of non-regular bounded modular operators on the hosting Hilbert C*-module; the supplied text does not state whether the conjecture is resolved.