Tautologicality conjecture for Lagrangian constant-cycle subvarieties

Let $X$ be a K3 surface and ${\mathsf M}$ a moduli space of stable sheaves on $X$ with special zero-cycle $c_{\mathsf M}$. A subvariety $V\subset {\mathsf M}$ is Lagrangian constant-cycle if it is Lagrangian and all of its points have the same class in the relevant rational-equivalence orbit for $c_{\mathsf M}$. Tautologicality conjecture. The class of any Lagrangian constant-cycle subvariety for $c_{\mathsf M}$ belongs to the tautological ring:

$$[V]\in R_{\star}({\mathsf M}).$$

The conjecture concerns the Chow classes of geometrically distinguished subvarieties and is motivated by known Lagrangian constant-cycle subvarieties and their associated Chow vanishing relations.