I have four plastic slider gadgets from Lakeland or Betterware though I expect you can buy them elsewhere. They are shallow dishes that you put the legs of chairs into and then push or pull across the carpet. May help?
I think the ply would work if you glued/screwed these underneath it so they slid on the carpet. They work really well. Get the biggest ones you can and if the ply is thick enough, choose ones that have a screwhole in the middle....you might need 5 per chair, 4 around where the circumference of the ring is, only on the other side from the chair IYSWIM, and gthen one in the centre. Amazon.co.uk User Recommendation