I don't know if I'd go down this path if I'm the team.
What are they going to do if they find out the answer is that he punched someone out in a bar or punched his wall at home? It's not going to bring him back any sooner. Do they want to take their relationship with Boozer to this negative of a place before he even steps on the court?
I can see wanting to know the truth, so that if there's other issues, you can go back and say "well look this isn't the first time", but you also have to realize that you're paying 75 million to this guy, and you don't want to have him set negatively against the organization prior to even playing a game. Otherwise you end up with a Ben Wallace.
A guy just going through the motions on a huge contract, untradeable and not helping you on the court. That's the sad reality of today's NBA where you can obligate players to play contractually, but you can't obligate them to care.
Granted, this is all hypothetical, because:
1: Boozer could simply be telling the truth (though if he is, their suspicion may cause problems)
2: If he's not telling the truth, he could be contrite, and work just as hard when he comes back
3: The Bulls might find out something is wrong and file it away rather than confront Boozer
4: It may be mandatory (or policy) to investigate all off court issues due to league insurance issues.
I know the truth brigade will come out and say we have to know! We have a right to know! I agree that the Bulls certainly have the right to know the truth if they haven't been told it, but that doesn't mean the truth won't cause them long term harm if they pursue it in the wrong way or do the wrong thing with it.
Tread lightly with the Boozer situation, because the rumors of him not giving full effort were already there from Utah. The concerns about the downside of this signing existed long before the signing was made, and while I completely agree with the move given the position the Bulls were left in, they need to protect their investment, and part of that protection might be just trusting Boozer on this one, even if he isn't telling the truth.