DepsGraph and Bone Constraints with no Influence

I’m having an issue with a (non-)cyclic dependency relating to Bone Constraints. Blender assumes there’s a cyclic dependency and/or treats it like there is even though the Bone Constraint causing the cyclic dependency’s Influence is set to 0. If I remove the Constraint, however, the cyclic dependency is no longer, naturally. Is there any possibility of making it so that Bone Constraints whose Influences are 0 are removed from the DepsGraph, similarly to if it was never there to begin with?

It seems you can make a driver for the limiter activation button

If you mean image right next to the Bone Constraint’s name, then unfortunately, the Rig still lags behind.