Let $C$ be a compact 2-dimensional cylinder $[0,1]\times S^1$. Let $A,A'$ be the two connected components of its boundary.

Let $Q$ be a square. Let $a,a'$ be a pair of opposite edges of $Q$.

Consider a smooth imbedding $f\colon Q\to C$ such that $$f(a)\subset A,\, \, f(a')\subset A'.$$ Assume moreover that $f(\partial Q\backslash (a\cup a'))$ in contained in the interior of $C$.

Question. Is it true that $C\backslash f(Q)$ is connected?

A reference will be helpful.