Hodge lift conjecture for GPS objects

Let $X$ be a holomorphic exact symplectic manifold with compact core, and let $L$ be a complex Lagrangian submanifold. The object $\operatorname{GPS}(L)$ denotes the microsheaf associated with $L$ by the Ganatra--Pardon--Shende construction.

Hodge lift conjecture. For any complex Lagrangian $L$, the object

$$\operatorname{GPS}(L)$$

has a lift to a Hodge microsheaf.

This is the Hodge-theoretic motivation for seeking Hodge versions of wrapped microsheaves. The source does not provide a general existence theorem, so the lift remains conjectural.