The surgery-contactomorphism conjecture for Legendrian knots in the standard 3-sphere

Let $L$ and $L'$ be Legendrian knots in the standard contact 3-sphere $(S^3,\xi_{st})$. Write $L(-1)$ and $L(+1)$ for the contact manifolds obtained by Legendrian $(-1)$- and $(+1)$-surgery on the knots. Surgery-contactomorphism conjecture. If $L(-1)$ is contactomorphic to $L'(-1)$ and $L(+1)$ is contactomorphic to $L'(+1)$, then $L$ is Legendrian isotopic to $L'$ in $(S^3,\xi_{st})$.