Conjecture on hyperbolic knots admitting Nil Seifert fibred surgery

Let $K$ be a hyperbolic knot in $S^3$. A surgery on $K$ is a Dehn surgery whose resulting manifold is a Nil Seifert fibred space. Nil surgery conjecture. If $K$ admits a surgery yielding a Nil Seifert fibred space, then $K$ is one of the four hyperbolic Berge knots given in parts (b), (c), and (e) of the Addendum. The preceding results identify the possible Nil Seifert fibred spaces and slopes for non-trefoil knots; the conjecture asserts that the four listed hyperbolic Berge knots exhaust the hyperbolic-knot cases.