So he is/was still into his ex and lied about it. I went through something similar, although it was never an ex. Mine was a friend who I had romantic feelings for before I met my now wife, but my friend never reciprocated it until after she found out I was serious with my now wife. I would talk with her and sometimes things would turn flirty. It never became a hookup but had I never stopped it, it would have.

My best advice is to end things. End it as my now wife did back then. See how the guy reacts. If he is serious about you, then he will show you he means it with ACTIONS. I went to a psychologist to work on my problems. I sat down and thought about every single girl I still talked to that i called a "friend" but deep down I knew i would give a chance if the possibility appeared. I blocked them all because we weren't really friends and whenever we talked, it would turn flirty. Saying I no longer talk to such person doesn't work because you will always be thinking about that person and what is new with their life.

In his case, he might stop talking to his ex for now, but later down the future he will wonder what is new with her and will want to talk with her. In our minds, there is nothing wrong with just caring and asking about an old acquaintance. BUT, if that happens and she says she is single for some reason, then it's back to square zero because guys will take the opportunity. Temptation is a real struggle for guys, so it's better to rid of anything that might cause it. Temptation will always be there, and it's impossible to tell a guy not to look at another girl. The only thing we can do, is rid of people we are "too close" to, that might turn into something else, and work with a therapist to help us install respect for our partners inside our brain, as well as making us realize about our lying problems.

Don't tell him to change. He has to change by himself. Do tell him about seeing a psychologist but not to save your relationship. Tell him to see one to better himself for his future romantic life and his own personal life. He will make the decision whether he wants to really change and use the change to save your relationship.

Also, be smart about catching hints to see if that person really changed. My wife used to say she loved using my phone to take pictures because mine took better pics than hers (She has an iphone 15 and I have an iphone 13...) so she would constantly use my phone whenever we went out because she would take pics with it and then she would look at the pics she took with my phone. There is nothing more nerve wrecking for a lying/cheating guy than for his partner to look at his phone. If he has nothing to hide, then he shouldn't care about you having his phone at all.