Yes, but there are rational reasons to continue believing something false after losing an argument about it. One example is believing that one's own arguments were inferior to what they could be and failed to present the real source of one's beliefs. Being bad at representing one's own beliefs is incredibly shameful and a good reason to feel insecure.
Additionally, your argument is based on a premise that isn't always true. When you say that leaving a discussion is about actively avoiding changing the fact that one is wrong. As I said before, often insecurity is the only reason someone suddenly leaves a discussion in which something they believe is being contested. Nothing stops a person from changing their mind AND being insecure about it. Sometimes a person doesn't even realize they've changed their mind until they look back in retrospect.
Is it any of your business what someone thinks inside their own head? If they are too insecure to keep talking about it this time then why not just wait until they are more secure and feel like their perspective has improved?
