Microsheaf wrapping conjecture for complex Lagrangians

Let $X$ be a holomorphic exact symplectic manifold with compact core, and let $\mu\operatorname{sh}_{\mathbb C\text{-}c}(X)$ be the finite-microstalk part of the triangulated closure of the perverse microsheaf $t$-structure. Let $L$ be a complex Lagrangian submanifold.

Microsheaf wrapping conjecture. There exists an inductive system $\{\mathcal E_i\}$ in $\mu\operatorname{sh}_{\mathbb C\text{-}c}(X)$ such that

$$\lim_{i\rightarrow +\infty}\mathcal E_i\cong \operatorname{GPS}(L).$$

The source explicitly remarks that this conjecture seems to be false in general and should instead be viewed as a guiding principle; thus the proposed statement is refuted as stated.