I've heard it said (e.g., in the accepted answer to this MO question) that a major obstacle to an effective theory of Ricci Flow in dimension 4 is the absence of the Hamilton-Ivey pinching phenomenon. I'm curious about the possibilities for such a pinching in dimension 4, but I couldn't locate any information about it. I'm curious about 2 complementary questions in this regard.

• Are there any known partial results or indications of what such a pinching may look like in dimension 4?
• Are there known examples that constrain the form of or throw doubt upon such a possible pinching?

Any thoughts or references to the literature are welcome.