The Simple Loop Conjecture for maps from surfaces to 3-manifolds

Let $S$ be an orientable surface, let $M$ be a 3-manifold, and let $f\colon S\to M$ be a map. The induced homomorphism is

$$f_*\colon \pi_1(S)\to\pi_1(M).$$

Simple Loop Conjecture. If $f_*$ is not injective, then some element of $\ker f_*$ is represented by a simple, that is, embedded, loop in $S$. This conjecture generalizes Dehn's lemma from inclusions of surfaces to arbitrary maps and has been verified for some classes of 3-manifolds, but remains open in general.