Greene's doubly primitive conjecture for knots in $S^1 \times S^2$

Let $K$ be a knot in $S^1 \times S^2$ and suppose that a longitudinal surgery on $K$ produces a lens space. Greene's conjecture. The knot $K$ is doubly primitive. This is the broader replacement for Greene's original Berge-Gabai-only conjecture and is presented as the main conjecture of the paper; the authors propose families intended to account for all such knots.