The simple loop conjecture for 3-manifold groups

Let $M$ be an orientable $3$-manifold, and let $\Sigma$ be a closed orientable surface. The simple loop conjecture. The kernel of every non-injective homomorphism from $\pi_1\Sigma$ to $\pi_1 M$ contains an element represented by an essential simple closed curve on $\Sigma$. This extends Gabai's simple loop conjecture from homomorphisms between surface groups to homomorphisms into fundamental groups of closed orientable $3$-manifolds. The statement is presented as a conjecture in the source; the paper instead gives counterexamples to a version with target $\operatorname{PSL}(2,\mathbb C)$, so its status here is left open.