Paracyclic lift conjecture for microlocal sheaf categories

Let $\Lambda\subseteq S^*M$ be either a swappable Legendrian stop or a full Legendrian stop, and let $m_\Lambda^l$ be the left adjoint of microlocalization. For each $n$, consider the simplicial $\infty$-category

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

Paracyclic lift conjecture. This simplicial $\infty$-category can be lifted to a paracyclic $\infty$-category.

The conjecture extends the geometric Waldhausen $S$-construction associated with semiorthogonal decompositions and spherical adjunctions. The supplied text gives no resolution, so the existence of the paracyclic enhancement remains open.