Any given company could stop training tomorrow, and, as some others have said here, they'd be generating quite a bit of profit until their models visibly fell behind, however long that ended up taking, at which point they'd probably just fall over completely.

Over the whole industry? No; they can never, ever stop training, or they'll cease to be useful at all very soon.

Training is what keeps the models up-to-date on current events, which includes new programming languages, frameworks, and techniques. It's already been observed that using LLM assistance on some types of programming is much more effective than others, based on how well-represented they are in the training data: if everyone stopped training tomorrow, and next month a new programming language came out, none of them would ever be able to help you program in that new language.

This can be extended to other aspects of programming, too. If training stopped, coding assistants would gradually start giving you wrong answers on how to implement code for APIs, frameworks, and languages that continued to evolve, as they will always do, in much subtler (and likely harder-to-debug) ways than how they'd deal with a new language whose existence they don't even know about.