Homology-class classification for longitudinal $S^1 \times S^2$ surgeries

Let $L$ be a lens space, let $[?]$ be a knot in $L$ representing $[?]$ and let $[?]$ denote the homology class of a Heegaard-solid-torus core. Homology-class conjecture. If the knot admits a longitudinal surgery to $S^1 \times S^2$, then, up to homeomorphism, $L=L(m^2,q)$ and its homology class is one of the classes listed in Theorem~. Together with the doubly primitive conjecture, this would make that theorem a complete list of such knots in lens spaces.