When working on your playing, you should always remember that you can go too far in any direction. The aperture can be TOO open. The slide can move TOO fast. Going along with that same thought, the corners can be TOO firm. For the most part, it is important to keep your corners firm and steady. The point here is to be careful not to be overly muscular and inflexible. Within the structure of having corners that don't collapse and turn into a smile in the upper register, there should be some degree of relaxation in the corners. If you have complete, unrelenting muscularity in your corners, the muscle contraction in your face will eventually cause a buildup of lactic acid which in turn will cause stiffness in your embouchure and a lack of endurance. The musculature needs to relax at some point to allow the lactic acid to cycle out of the embouchure. This is why playing soft or playing low can help you get your chops back when you are feeling tired. When playing soft or low, your corners don't have to be quite as firm to support the air stream appropriately. It's also the reason why a healthy vibrato can help your chops feel more flexible. That motion helps the acid cycle through the musculature and not build up in a static environment.
An exercise I like to try with students working on the upper register and corner construction is to have them play an F in the bass clef staff. Then work up a major scale as far as possible while attempting to keep the corners in the exact same place as they were when the scale began. The key is to not worry about squeezing the corners as hard as possible or making them super firm or thinking not to smile. The goal is to simply have the corners remain in the early position with minimal stress. Maintain a sound that is relaxed and flexible.
To the students... Please don't tell people that Weston Sprott said to play with a lack of structure or firmness in your corners. I am simply making the point that the idea of firmness in the corners can go too far, sometimes to the point that a lack of flexibility and endurance result. I definitely agree with the idea of a structured embouchure setup where the corners don't spread or collapse in any register.