Waldhausen S-construction conjecture for microlocalization

Let $M$ be the ambient manifold, let $\Lambda$ be the Legendrian used to define the microlocalization functor, and let $m_\Lambda^l$ denote its left adjoint. Write $T_{j\epsilon}(\Lambda)$ for the corresponding translated Legendrian stop and $\operatorname{Sh}_{\bigcup_{j=0}^{n}T_{j\epsilon}(\Lambda)}(M)$ for the category of sheaves on $M$ with singular support in that union.

Waldhausen S-construction conjecture. The relative Waldhausen $S$-construction of $m_\Lambda^l$ is the simplicial $\infty$-category

$$S_n(m_\Lambda^l)=\operatorname{Sh}_{\bigcup_{j=0}^{n}T_{j\epsilon}(\Lambda)}(M).$$

This identifies the relative Waldhausen construction arising from the semiorthogonal gluing with a geometric category of sheaves. It is proposed in the context of $K$-theory and the simplicial $\infty$-categorical formalism of Dyckerhoff, Kapranov, Schechtman, and Soibelman.