It isn't the foam tyres that cause the problem, it's the rubber tyres! The problem is the oil used as a tyre treatment for rubber tyres, and then the bits of rubber that get into the carpet. All they need is a nice clean rubber tyre to come along and they glue themselves to it. As they get a hold on the tyre, they form an even better surface for more to get glued on!
Foam tyres do exactly the same thing on Ardent's carpet, and any other carpet running rubber tyres. However, foam tyres do no such thing on a carpet that only runs foam tyres. Sorry to burst your bubble, but if you want to avoid the problem you restrict the carpet to foams only until the race!! HTH
Team Doris second your experience of the tyres - both kit and Sorex 28s.