Thanks for the feedback, gbenga.animator.
It just needed to start a frame earlier to “constrain”.
I would always use the binding (BD v nb) joints, e.g., R_ or L_BD_FK_Finger_Mid_jnt_2, which has the Priority Expression 60.
At one point, you will commit the rig, and press the c key with the active Character Object, then the best is to use the binding joints anyway, from my point of view.
Is that helping?
If nothing helps and your animation is finalized, you could always use Cappucino to get an object that is constrained baked. If there is any delay, the keyframes are easily moved to the left to get things in sync. But, yes, that is more of a hack than how it should work.
All the best