Will existing Activision account progress auto-pop Steam achievements?

I have a question for anyone who's played Call of Duty on both BattleNet and Steam.

I already own Cold War and Black Ops 6 on Battle.net. I've completed a lot of the content that would be tied to Steam achievements. I am thinking about buying both again on steam to complete my black ops and zombies achievement collection (DAMN YOU BLACK OPS 4), but I want to know if I link the same Activision account to Steam will the achievements automatically pop on the ones I have already done? or will I need to replay the achievements? or will I be permanently locked out of achievements entirely since I already have them on that activision account and should I make a new activision account for steam specifically?

Thanks!