The rank versus Heegaard genus conjecture: It states that given a closed (compact works as well, I think) hyperbolic 3-manifold $M$, the conjecture states that the Heegaard genus of $M$ is equal to the rank of $\pi_1(M)$. It's is relatively easy to see that the Heegaard genus is always greater than or equal to the rank just by looking at the definition of a Heegaard splitting, but the other inequality is not known.
For non-hyperbolic 3-manifolds, there are examples where the genus is strictly greater than the rank (I don't have a reference for this off the top of my head).
