Integer surgeries rational homology cobordant to lens spaces

The Cyclic Surgery Theorem and Moser's work on surgeries on torus knots imply that for any non-trivial knot in $S^3$, there are at most two integer surgeries that produce a lens space. This paper investigates how many positive integer surgeries on a given knot in $S^3$ can produce a manifold rational homology cobordant to a lens space. Tools include Greene and McCoy's work on changemaker lattices which come from Heegaard Floer $d$-invariants, and Aceto-Celoria-Park's work on rational cobordisms and integral homology which is based on Lisca's work on lens spaces.