Even if you don't trust police at all, it makes no sense for them to lie about this. The truth of the matter will come out in court, lying has no upside and significant downside.
What is the downside of the police lying? Assuming they didn't have a warrant then they either lie and it doesn't make the situation worse as police are allowed to lie or they don't lie and guarantee a lawsuit.
I'd understand the argument for why they're incentivized to get a warrant in the first place (even assuming parallel construction is a viable alternative to getting a warrant as they may be caught) but that doesn't mean they wouldn't lie after they got caught screwing up.
There is, honestly, less of a reason to believe it's false. They may be lying but our legal system relies on us trusting the police to be honest and, ideally, strongly punishes those who abuse that trust.
I'm happy to concede that policing in America has long eroded that trust but it's still a requirement that you obey and openly communicate with anyone identifying themselves as law enforcement.
Actually yea - I was wrongish on this one. It's a state level decision[1] with different requirements and generally a requirement of suspicion of guilt. I was mostly thinking of surface level information (identification and the like) rather than any sort of detailed information that might contribute to guilt - but even that surface level information is still protected information.
