The Berge conjecture in grid-number-one form

Let $L(p,q)$ be a lens space and let $K\subset L(p,q)$ be a knot. A knot has grid number one if it can be represented by a grid diagram of grid number one.

Berge conjecture. If surgery on $K\subset L(p,q)$ yields the three-sphere, then $K$ has grid number one.

This is presented as an equivalent form of the Berge conjecture and is the formulation targeted by the paper's combinatorial methods. Its status is not specified in the source.