Can me and my ex still work out?

I’m reposting this as it was taken down.
Me (F 29) and my ex (M 30) were together for 6 years. We broke
up last year not on the best of terms because of a boundary that I crossed and he didn’t appreciate. He thought I cheated on him and he made me get a
lie detector test which I passed. After that I didn’t want to be with him, I actually did but I felt like I needed time. After 4 months I started talking to
someone new and started a relationship too quickly and he was deeply hurt. I realized I still wanted to be with him and I broke up with the new guy. But now
my ex of 6 years doesn’t want to make things work. He is back and forth and says he wants to be with
me still but then says he doesn’t. I am heartbroken but I also understand why. His family and friends wouldn’t approve of us getting back together
because he did go through a rough time after our breakup. Now I am not sure of what to do. I want to try really hard and try and make things work. Or should I just let the relationship go since there is so much hurt between us? He did try very hard during the relationship to make it work and I was a little careless during the relationship. I didn’t have the
best communication with him specially during our arguments. But I am going to therapy now to unlearn some of my unhealthy habits.

TLDR: should I let go of a relationship of 6 years or should I fight for the relationship back?