  Let $X$ be a smooth manifold and $\mathbf{k}$ be a commutative (or at least
$\mathbb{E}_2$) ring spectrum. Given a smooth exact Lagrangian
$L\hookrightarrow T^*X$, the microlocal sheaf theory (following
Kashiwara--Schapira) naturally assigns a locally constant sheaf of categories
on $L$ with fiber equivalent to the category of $\mathbf{k}$-spectra
$\mathrm{Mod}(\mathbf{k})$. We show that the classifying map for the local
system of categories factors through the stable Gauss map $L\rightarrow U/O$
and the delooping of the $J$-homomorphism $U/O\rightarrow
B\mathrm{Pic}(\mathbf{S})$. As an application, combining with previous results
of Guillermou [Gui], we recover a result of Abouzaid--Kragh [AbKr] on the
triviality of the composition $L\rightarrow U/O\rightarrow
B\mathrm{Pic}(\mathbf{S})$, when $L$ is in addition compact.
