First, were we to lie in our privacy policy, we would be subject to GDPR fines of up to 20 million Euros, since we have both European customers, and a presence in the EU.
Second, there has already been a court case. We were ordered by a Swiss court to hand over logs, and we stated truthfully (under penalty of perjury) that we did not have the logs requested. This case was previously disclosed here: https://protonvpn.com/blog/transparency-report/
I'm not terribly well-versed in the international (or Swiss) legal system but are portions of that request public record, or would it be possible to put portions of it online, verbatim?
It would really strengthen the case to your customers because whilst claiming you had a request when you didn't isn't illegal, falsifying court documents definitely is.
I realized another way that would work for you guys (but is out of your hands) is fighting a court case about this. You'd be legally compelled to tell the truth and very screwed if you deny but then it comes out there is logging or mining going on. It's not ironclad but it is how most VPNs end up being considered 'solid'.