I found an interesting conjecture in the paper "Hyperbolic volume, Heegaard genus and ranks of groups" by Peter B. Shalen.
It says that if $M$ is a compact, orientable, hyperbolic 3-manifold, then the rank of $\pi_1(M)$ is equal to the Heegard genus of $M$.
How open is this conjecture? Do you know how to prove it or can you find a counterexample?