Bounded lensbordant surgeries for knots with nonzero \nu^+

Let $K$ be a knot in $S^3$. A lensbordant surgery on $K$ is a positive integer surgery that is smoothly rational homology cobordant to a lens space, and let $u^+(K)$ denote the knot invariant used in the statement. Bounded lensbordant surgery conjecture. There \exists a positive integer $N$ such that, for every knot $K$ in $S^3$ satisfying $u^+(K)\neq 0$, the number of lensbordant surgeries on $K$ is less than $N$. This is an analogue of the Cyclic Surgery Theorem, replacing lens-space surgeries by surgeries whose results are smoothly rational homology cobordant to lens spaces; the conjecture asserts a uniform bound independent of the knot.