Issue in exported animation with specific rig from Maya

Hello guys,
Im trying to move a specific rig from Maya to Blender. I would like to kindly ask experienced riggers and animators for an advice what can be the issue.

The rig has these requirements:
Specific mesh object must control the joint to deform other meshes. The solution have to meet this requirement, ideally less rework the better :slight_smile:

I ve moved the animation and the rig to the Blender and it works…almost. But there is a strange behavior of vertices, that Im not able to figure out. Skinweights on vertices are identical, so these are exported well. Could that be that transform of parented boxes are wrongly transfered to bones in the constraints?

Here I am adding gifs to describe the problem.
Thanks in advance

This is the simple animation. Boxes are controlling bones to deform the plane.

In Blender I am using constraints on bones to meet requirements. Boxes are controlling bones to deform the plane. But there is a small decay of middle vertices on the plan as you can see on gif.

Here I muted the animation of the box, so you can clearly see what is the issue.

Im also adding blend file.

testExportAnimation.blend (602.7 KB)