Berge's conjecture on lens space knots

Let $K\subset S^3$ be a lens space knot, meaning that some positive integer surgery on $K$ produces a lens space. Berge's construction gives a family of such knots. Berge conjecture. If an integer surgery along a knot $K$ in $S^3$ produces a lens space, then $K$ must arise from Berge's construction. This conjecture seeks to classify all knots admitting lens space surgery; it is presented here as an open problem.