If it's a private benchmark, how would they know it's a benchmark?
Look up eval awareness. There are all sorts of subtle signals an LLM could use to know if something is shaped like a benchmark or not. But that's not a bad thing, every animal behaves differently in play vs real-life.
Yes the model can do that, but it would not be "trivial" (as GGP had said) for Anthropic to opt out accounts performing private benchmarks from this theoretical model downgrade people think is happening.
The most plausible approach would be for Anthropic to train some separate classifiers to detect benchmarks. I'm not saying it's completely impossible but IMO this is conspiracy theory levels of effort.
In the context of public relations they would not need to ID every benchmark only the most credible or most public benchmarks many of which are openly available on github and whose problem-set could be fingerprinted.
You're absolutely right that it is a game of whack-a-mole and eventually we may have leading private + credible + popular benchmarks that ultimately dissuade the behavior, if it can be dissuaded.