Rank versus Heegaard genus conjecture for hyperbolic 3-manifolds

Let $M$ be an orientable finite-volume hyperbolic $3$-manifold. Its rank is the minimal number of generators of $\pi_1(M)$, and its Heegaard genus is the minimal genus of a Heegaard surface for $M$. Rank versus Heegaard genus conjecture. The rank of $M$ equals its Heegaard genus. This is a longstanding question in $3$-dimensional topology; the source notes that, if it is false, a related question about rank gradient has a negative answer.