Why would someone have to lie? They can just say "We can't comment on that" without providing an answer. And then customers can go "sounds pretty suspicious, time to switch VPN services".
The point of the canary, is to turn any non-answer into a practical, or at least a tentative, "yes". If you no longer see an explicit "no", it means "yes".
The difference between a canary and a "no comment" is that "no comment" is an extremely common thing to say whether an allegation is true or not so it's not very suspicious, while stopping a canary is very suspicious.
So it's like the scenario you outlined earlier, but more effective.
I'd hire a lawyer that knows how to deal with them and knows how to say no comment.
I've always felt that the warrant canary is a nerd's gotcha designed to get out of a sketchy legal process (NSLs) and that judges would be very unsympathetic. But IANAL.
At least from my perspective, it is not just a "gotcha" problem or whatever feelings it conjures up in a judge's mind.
When a company, who already has a canary in place, receives this kind of warrants, what _can_ the company practically do to comply with non-disclosure? It seems that lying is now the only option left, if the company must explicitly post a "no, we didn't receive such a warrant".