The contact cosmetic surgery conjecture for Legendrian knots

Let $(Y,si)$ be an integer homology sphere, and let a Legendrian knot be a knot represented by a Legendrian embedding in $(Y,si)$. Two contact surgeries on the knot with distinct smooth coefficients are cosmetic if the resulting contact manifolds are contactomorphic.

Contact cosmetic surgery conjecture. Any Legendrian knot in $(Y,si)$ that is not smoothly an unknot admits no cosmetic contact surgeries.

The paper establishes this claim for non-trivial Legendrian knots in integer homology sphere $L$-spaces, with a possible exception consisting of Lagrangian slice knots. The supplied text does not establish the conjecture in the full class of integer homology spheres.