We call the image of a smooth embedding $f:\coprod_{i=1}^n [0,1]\times\{i\} \hookrightarrow \mathbb{D}^2 \times [0,1] $ a string if $f(0,i)$ and $f(1,i)$ have the second coordinates $0$ and $1$ respectively, and also, $f$ sends interior to the interior. Let us denote the complement of a tubular neighborhood of a string by $X_f$.
Do the higher homotopy groups of $X_f$ vanish? I tried to prove that by showing that its universal covering is contractible, but I am not sure how it looks like.
Also, it should be closely related to the homotopy groups and the covering of a link complement in $\mathbb{S}^3$ but I am not sure either but also interested.