Question on consenting to a vehicle search

I saw one video where the officer told the suspect they had a warrant for their truck and were going to search it. Of course it would be easier if they had the key otherwise they’d have to tow it and find another way to get in (Question: do the police or the suspect pay for that?). So the suspect gives then the key.

Let’s consider the likelihood that the cops are lying and there was no warrant. By him handing the key in response to the lie, did he give consent for his truck to be searched?

Doesn’t look like it.

The majority opinion noted that the police could not rely on the grandmother’s consent inasmuch as it was given only after law enforcement claimed to have a warrant