The rank versus genus conjecture for knot exteriors

Let $K$ be a knot, let $E(K)$ be its exterior, and let $t(K)$ be its tunnel number. Write $rank(\pi_1(E(K)))$ for the minimum number of generators of the fundamental group.

Rank versus genus conjecture. It holds that

$$rank(\pi_1(E(K)))=t(K)+1.$$

This is the knot-exterior case of the rank-versus-genus problem, which the source describes as unsolved.