Suppose that $M^n$, $V^{n+2}$ are connected, compact smooth manifolds.

Let $f\colon M^n\to V^{n+2}$ is a smooth embedding.

Let $K_f$ be the kernel of the inclusion induced homomorphism $\pi_1(V-f(M))\to \pi_1(V)$.

Choose a meridian $\mu\in \pi_1(V-f(M))$.

Proposition 1.3 of Smith's paper, Complements of codimension two sub-manifolds I: The Fundamental group, Illinois J. Math. 1978 p. 232-239 says that if $f_*\colon \pi_1(M)\to \pi_1(V)$ is a surjection, then $K_f$ is normally generated by $\mu$.

Question: Is the condition that $f_*\colon \pi_1(M)\to \pi_1(V)$ is surjective necessary? That is, is there an embedding $f\colon M\to V$ such that $K_f$ is not normally generated by just one meridian?