The solid-torus realization conjecture for lens space surgeries

Let $S^1\times D^2$ be the solid torus and let $S^1\times S^2$ be the product of the circle with the 2-sphere. An integer lens space surgery is integer surgery on a knot in $S^1\times S^2$ producing a lens space; an integer $S^1\times D^2$ surgery is integer surgery on a knot in the solid torus producing $S^1\times D^2$. The solid-torus realization conjecture. If a knot in $S^1\times S^2$ admits an integer lens space surgery, then it arises from a knot in $S^1\times D^2$ with an integer $S^1\times D^2$ surgery. This is an analogue of the Berge conjecture motivated by the relation between lens spaces bounding rational balls and surgeries on knots in the solid torus; its general validity remains open.