If they do fake HDR and claimed otherwise, I hope they get sued.
There's no way to verify it because there's no HDR content on PS4. Probably the reason they got away with it.
Not sure what you mean by that.
I have an OG PS4 and it does "HDR" on the TV's I've mentioned such as the one of the best p/p TV's in 2018 such as the X900F.
The darks were darker and there highlights were slightly better in HDR mode BUT as you mentioned, it could be a "fake" form of HDR where perhaps the result basically equates to something like SDR with tweaked black levels and/or colors.
Someone with proper software and equipment could do a proper comparison with the same TV using the OG PS4 vs other models/competing consoles.
Vincent with PS4 Pro with HDR head to head to head using 3 Sony TVs:
Sadly, my impression of HDR and DV (even in movies) was quite sub par specially with the terrible blooming. Not sure if I'd be frustrated with OLED's ABL as some say that's where HDR can quite shine too due to high contrast though that makes me wonder on how much impact nits actually have since OLEDs can't even produce as high peak brightness.
Do forgive me if you were saying that the PS4 doesn't have HDR/DV content for movies. No idea about that currently.