Hopefully some pheasant experts will post on your problem. A couple things I read about are infectious sinusitis from MG or mycoplasma, and lymphoma tumors from Mareks disease which can affect pheasants. Here is the article to read...
Have you also posted a thread on the pheasant thread? Could the pheasant have injured the forehead and rubbed the beak in some wire fencing or chickenwire? Is that yellow pus or secretions on the beak or nostril? What is your general location? I am more familiar with chickens. To post on the...