Large filing cabinet recovery

Hi,
I have a tall 6 drawer filing cabinet that was stored in a farm shed for a while. The plyboard side panels have started delaminating and I want to replace them. I put this job aside thinking it needed pulling apart just to insert new panels. Now I'm looking at doing it a different way. Below you can see the worst panel was removed. I was looking at cutting new panels to the correct size but then cutting each panel into 3 pieces. Then I would individually insert a piece by bending so the top and bottom could be moved into the slots in the pine wood frame. Repeat for the other side. Then the centre piece. Finally I would cut another plyboard that fits neatly on the outside of this new piece but between the frame faces that are next to the slots. This would be glued against the already inserted faces to basically make a thicker plyboard.

My thinking was this would give it the structural strength close to what it had before.

Is there something wrong with this idea. Is there a smarter way of dong this?
Cheers Nick

1. Front of filing cabinet
2. Side of filing cabinet
3. Slot in frame where plyboard was removed

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