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Let 
$\{\Lambda ^\infty _t\}$ be an isotopy of Legendrians (possibly singular) in a unit cosphere bundle 
$S^*M$ that arise as slices of a singular Legendrian 
$\Lambda _I^\infty \subset S^*M \times T^*I$. Let 
$\mathcal {C}_t = Sh(M, \Lambda ^\infty _t)$ be the differential graded derived category of constructible sheaves on 
$M$ with singular support at infinity contained in 
$\Lambda ^\infty _t$. We prove that if the isotopy of Legendrians embeds into an isotopy of Liouville hypersurfaces, then the family of categories 
$\{\mathcal {C}_t\}$ is constant in 
$t$.
