The geometric presentation conjecture for orientable 3-manifold groups

Let $M$ be a compact, orientable 3-manifold. A geometric word in a free group $F(n)$ is a word that can be represented by a simple closed curve on the boundary of a genus $n$ handlebody. A 1-relator presentation of a group is a presentation of the form $\langle x_1,\ldots,x_n\mid w\rangle$.

Geometric presentation conjecture. The group $\langle x_1,\ldots,x_n\mid w\rangle$ is the fundamental group of an orientable 3-manifold if and only if $w$ is a geometric word.

The source also formulates this through the equivalent claim that every 1-relator presentation of $\pi_1 M$ is geometric, and adds that, when $M$ has no 2-sphere boundary components, every such presentation is induced by a Heegaard splitting of $M$. The conjecture removes the fibered-manifold hypothesis from the preceding theorem and concerns which 1-relator groups arise as orientable 3-manifold groups.