Will My Achievements Also Count In Steam If I Earn Them In EA App

Hello. I want to purchase Battlefield 2042 and another game (both of which have the same achievements on Steam as well as the EA App). Steam for some reason is giving me really high prices, whereas EA App is giving me way lower prices in my local country's currency. So it's obvious to buy from the EA App...

But I want my achievements to appear in my Steam profile, as I don't use my EA Profile other than saving all the cloud data, Steam is what I use the most.

Last time I purchased NFS Heat from Steam and all of my achievements were also covered in the EA App. But Can anyone please tell me if this would work the other way around as well... getting an achievement on EA App would also give me the achievement on Steam (if we buy the game on EA App and not on Steam)?

Another similar thing I was considering was to buy EA Pro which is almost 10 times cheaper in the EA Store (EA App) relative to Steam. I think Steam only gives the prices in USD whereas EA Store is giving me prices in my local currency, hence it gives way cheaper prices. So my question is similar here as well, if I buy EA Play or Pro (whichever is better) on the EA Store, then would my game achievements on those games also appear in my Steam Profile (because I don't have them in my library on Steam)?

It is really confusing that if we buy anything on Steam and it is compatible with logging in via an EA Account, then it's a happy ending as both accounts are running and updating everything between them. But if we only buy something from the EA Store, then it remains only in the EA Store (if it is as I understand it)

I'm really sorry that I got into a lot of details, but I just wanted a clear explanation of how this system actually works so I could purchase all my future games keeping this in mind. It would be really appreciated if someone could please answer this quickly as only one week is left before the sale ends on those BF games. Thanks 🙂