If $$E$$ is a Banach space, denote with $$I$$ the identity map of $$E$$, denote the space of continuous functions from $$E$$ to $$E$$ with $$\operatorname{C}(E,E)$$ and denote the space of homeomorphism from $$E$$ onto itself with $$\operatorname{Homeo}(E)$$.

Definition 1: If $$E$$ is a Banach space, $$\emptyset\neq A\subset K\subset E$$ and $$O\subset E$$, we say that $$O$$ strongly links $$(K,A)$$ if $$\forall\varphi\in \operatorname{C}(E,E), (\varphi|_A=I|_A)\implies (\varphi(K)\cap O\neq \emptyset).$$

And

Definition 2: If $$E$$ is a Banach space, $$\emptyset\neq A\subset K\subset E$$ and $$O\subset E$$, we say that $$O$$ weakly links $$(K,A)$$ if $$\forall\varphi\in \operatorname{Homeo}(E,E), (\varphi|_A=I|_A)\implies (\varphi(K)\cap O\neq \emptyset).$$

Clearly, if $$O$$ strongly links $$(K,A)$$ then $$O$$ weakly links $$(K,A)$$.

Main question: If $$O$$ weakly links $$(K,A)$$, is it also true that $$O$$ strongly links $$(K,A)$$?

A secondary question: if the answer to the main question is no, are there any sensible topological properties (e.g. compactness, connectedness, being the closure of an open set, being $$A$$ the boundary of $$K$$...) that we can impose on $$A$$ or $$K$$ or $$O$$ in order to get the equivalence of the two definitions?

Comments: In order to avoid pathologies, I set the question in the context of Banach spaces, where I need it in the context of generalization of theorems of mountain pass type.