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.
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.