Formal isotopy detection by contact push-offs

Let $n$ be a positive integer, and let $(S^{2n+1},\xi_\text{st})$ be the standard contact sphere. Let $\Lambda_0,\Lambda_1\subseteq(S^{2n+1},\xi_\text{st})$ be smooth Legendrian submanifolds that are formally Legendrian isotopic, meaning isotopic through formally Legendrian embeddings. For a Legendrian submanifold $\Lambda$, let $\tau(\Lambda)$ denote its contact push-off, obtained from the contact submanifold

$$\tau(\Lambda)=(\partial(T^*\Lambda)\times\{0\},\xi_\text{st})$$

under a contactomorphism from a neighborhood of $\Lambda$ to $J^1\Lambda$. Contact push-off conjecture. If $\tau(\Lambda_0)$ and $\tau(\Lambda_1)$ are contact isotopic, then $\Lambda_0$ and $\Lambda_1$ are Legendrian isotopic. The conjecture asks whether the contact push-off retains enough information to detect the Legendrian isotopy class among formally Legendrian isotopic submanifolds; its status is not resolved in the supplied source.