The whole “lower pressure for more grip” thing is a generalization, and doesn’t account for the car’s behaviour at various stages of the corner. Upping front pressure will increase steering/turn-in response at the expense of the front tire breaking away/giving up grip more suddenly. That is, you are trading progressive/delayed grip loss for increased response. The advantage of higher front pressure is only for the entry stage of a corner. The trade off is less mid corner bite.
Similarly, dropping pressure will delay grip loss (to a point–because too low will just speed up grip loss also) at the expense of response. Car will be more sluggish in response to driver input but take longer to break away when the traction limit is reached. This is opposite to the above. A lower front pressure makes the car understeer more AND eat more tires at the entry stage of a corner. However, you will have more mid corner front-end bite.
The rear is a bit more complicated, as it depends on how you drive, and whether or not you throttle steer, trail brake and how you like to make corner corrections to get rotation. Without knowing those details, the rear is simply “lower psi for more grip”. I’d go down to as low as 34psi on a street tire. No lower though.
Every tire is different, and there is a proper way to do tire pressure (which I do not do - it takes a lot of patience, recording data on an excel sheet and varies with ground temps). That said, I found best results when I ran 39.5front, 40.5rear hot pressures. I set the rear higher than the front to encourage the back end to break away more swiftly for more rotation when the car was stock. Pressures don’t have to be so high if your sidewalls are stiff. Sadly, continental sidewalls are marshmellows that need all the help they can get! Same with PSS.