The US and EU have reported broadly similar inflation numbers for the last five years or so, with the EU generally being a little lower. If your thing was true, you'd expect a vast gulf in prices to have opened up, but it doesn't really seem to have.

Anecdotally, I was in SF once or twice a year for a few years before the pandemic, and it felt like mostly similar prices to Dublin (an expensive European city). I've also been back a few times since the pandemic, and it now feels somewhat more expensive than Dublin, but not dramatically. Which is about what you'd expect if the official inflation figures were broadly correct. Obviously this is super-anecdotal, mind you.