Sheaf Quantization of Legendrian Isotopy

Let $\{{\mathrm{\Lambda }}_t^{\mathrm{\infty }}\}$ be an isotopy of Legendrians (possibly singular) in a unit cosphere bundle $S^{\ast }M$. Let $Sh(M,{\mathrm{\Lambda }}_t^{\mathrm{\infty }})$ be the differential graded (dg) derived category of constructible sheaves on $M$ with singular support at infinity contained in ${\mathrm{\Lambda }}_t^{\mathrm{\infty }}$. We prove that if the isotopy of Legendrians embeds into an isotopy of Weinstein hypersurfaces, then the categories $Sh(M,{\mathrm{\Lambda }}_t^{\mathrm{\infty }})$ are invariant.