My surgeon was strongly confident on my ditching the brace at 7 weeks with no sleeve or other support. I am 66 so that didn't hold him back. He was much more conservative on the running saying that puts great pressure on the tendon. He wanted me to wait for a year on that, but he was focused on running races. Anyway, going without the brace really helped me get stronger.
I have seen other protocols calling for a short transition period using a sleeve. I didn't need it but I would follow your medical/pt advice and your own instincts. I did use the sleeve when I first tried to run.