here it is in easy to follow steps, I hope
1. materials needed- furniture finishing steel wool pad #0000. nothing coarser will do. "Howard" restore-a-finish, color-nuetral. "Howard" feed & wax, its a mixture of orange oil and beeswax. pads of cotton cloth and paper towels. the Howard products are available in any good store that carries furniture refinishing products in USA. Don't know about Europe but web site is WWW.howardproducts.com
2. wet, don't soak a cotton pad about 3" square with the neutral restorer (wear nitrile or latex gloves) and apply to the surface and rub it in going with the grain. use moderate pressure and do about 2 rows at a time. don't worry about the restorer leaking in around the leds. you only use enough to wet the board not soak it. turn the pad often to a clean side. I use about 4or5 layers for a pad. you will see all the crud that is removed as you go. when done, completely dry the surface with paper towels.
3. rewet the board with a clean pad and lightly buff about 2 rows at a time with the steel wool. DO NOT press hard, just very lightly buff back and forth relativly quickly always with the grain. dry when done with paper towels and check the circle marks. some may still be showing and need a repeat of step 2. one board took me 4 repeats.
4. the board should now have a uniform dullish look to it . not to worry, apply and rub in vigoruosly a coat of the oil&beeswax with the grain and let dry 5 to 10 min. Lightly buff soft cotton or flannel cloth. I usually put 3 applications of beeswax&oil to obtain a nice satin finish.
5. The boarders may be done in the same manner so you don't have a satin playing field with shiny boarders. Don't worry about the lettering, it won't come off as its under the finish
6. my theory is that these marks are not caused by sitting in sunlight as I have read. if that were the case you would only develope these marks where the pieces stood in their starting positions most likely. I think its a reaction between the finish that is used by the maker and vapors from the glue used to hold the felts on the pieces. how else would you get these marks on all 64 squares. it probably starts when new and everything is not completly cured and doesn't take much to start the reaction. the problem is the finish not the wood in my experience with this problem. good luck to anyone who wants to try this. I swear by it and feel free to asy any questions. there, all done.