The co-rank conjecture for 3-manifold groups

Let $M$ be a $3$-manifold. Its co-rank $c_{1}(M)$ is the maximal rank of a free group homomorphically surjected by $d70_1(M)$, and $b_{1}(M)$ is its first Betti number.

Co-rank conjecture.

$$c_{1}(M) \geq \frac{b_{1}(M)}{3}.$$

For compact $3$-manifolds, the co-rank is also called the cut number. The conjecture is attributed in the source to J. Stallings, with origins in work of T. Kerler connected to quantum invariants; the paper gives explicit closed and non-compact finite-volume hyperbolic counterexamples.