The simple loop conjecture for 3-manifolds

Let $\Sigma$ be a closed surface and let $M$ be a closed 3-manifold. Suppose that $F:\Sigma\to M$ is a 2-sided immersion, with induced map

$$F_*:\pi_1\Sigma\to\pi_1M.$$

Simple loop conjecture. If $F_*$ is not injective, then there is an essential simple loop in $\Sigma$ that represents an element of the kernel of $F_*$. This generalizes the Loop Theorem to immersed surfaces. The conjecture is proved in the source for closed 3-manifolds modeled on $\mathit{Sol}$; the statement in full generality is not resolved here.