at.algebraic topology – Embedding with vanishing images of homotopy groups

Let $f$ be a locally flat embedding from $S^2 times mathbb R^2$ to $S^2 times mathbb R^2$ such that $f_*(pi_k(S^2 times mathbb R^2))=0$ for any $k ge 2$.

Can we find a domain $U$ that contains the image of $f$ such that $U$ is homeomorphic to $mathbb R^4$?