Hi Saverio, I don't know how old your son is, but in any case, as a private student myself, I understand the issue and, in fact, this year I will receive my diploma, God willing, so I find myself in roughly the same situation as your son. Emotion depends a lot on a person's character and how it was formed during childhood. I can tell you my story, but obviously it should be viewed personally rather than as a MUST to follow. I was educated to overcome shame, and often emotion is a corollary of shame. I remember, in fact, when I was a child walking with my father and I was thirsty. My father wouldn't go to the bar to get me something to drink, but would accompany me outside the bar and tell me to go in and ask the bartender for a glass of water. Obviously, this is nonsense, but every age has its problems and fears to overcome; it's not that this story solves the problem, but it's just to let you understand how, in my case, shame was overcome right from the start. Regarding the issue of emotion, then, this is also something to overcome, and in my opinion, victory can be achieved essentially in 2 ways. The first method, which in my opinion is the most difficult, is psychological self-persuasion; let me explain better. If your son feels prepared and the pieces for the 5th [exam] come out as they should, he must convince himself that he has nothing to fear and that at most, as a private student, he might be put under pressure over some triviality, but that this will in no way result in his failure—at most, a lower grade. Furthermore, one must be aware of the fact that emotion can only make things worse and therefore make you play "worse" than you do at home, precisely because you lose concentration and instead of focusing your attention on the piece, you start thinking with your brain about things like: "Oh my god, what will happen if I make a mistake?" This puts you into a vortex where, by thinking about it too much, fear sets in; fear makes your hands shake and so, in the best-case scenario, you commit the error and in the worst-case scenario, you freeze. So the mental process is as follows: I have studied -> I have nothing to fear -> I will do my best -> I will enjoy my exam. I'll pause for a moment on this last step: "I will enjoy the exam."
You go to exams to play; it doesn't matter if there is a commission of 3, 4, 5, 10, or 1000 teachers in front of you, you are there with the piano—the moment you start, it is just you and it! I say to your son: "Think about the music, live it for the pleasure of living and play it for the pleasure of playing; don't think about the exam! The exam is secondary to the primary thing, which is the music, and that's it."
The second method, instead, is what I call "face-breaking," ehehe.
Have your son play in front of people as much as possible; the more the better, the more tense the atmosphere is, the better he will break his face [get used to it].
I still remember when I did my first piano recital, my heart was racing at 1000! But by doing recitals, competitions, exams, and concerts, you eventually stop thinking about it. Of course, a tiny bit of emotion is always there before starting, but it must disappear when you sit down in front of the piano.
I realize that time is running short, so if I may, I advise you to put these 2 "methods," if we want to call them that, into practice; they worked for me. Talk to your son and put him in a state of serenity; tell him that if he doesn't pass, nothing will happen (which is perfectly true) and that the moment he places his hands on the piano at the exam, there will only be him and the piano. The piano must be lived, and you must treat it and feel it as if it were your only true friend who understands you, who gets inside you and manages to bring out all the most hidden things within you—in your brain and in your heart—that words could never bring out because they are not as strong as music.
I'll add one last note to this last concept. Philosophers themselves have fought to have music recognized as "supreme art"; in effect, religions, atheism, or beliefs aside, music, in my opinion, truly brings you closer to something higher—let's call it God, let's call it the Universe, let's call it Allah.
I hope I have been useful to you; for any questions we are here, do not hesitate to ask us.
Best regards,