"You’re not really Jewish", is this going to be a thing?

If I convert, I know it will likely happen through Conservative Judaism (nothing is for certain yet) for many reasons I won’t get into here. And while I’m aware that I have to accept some level of rejection from some Jewish people who will never see me as “really Jewish”, I’m concerned with this whole “your conversion wasn’t right, and therefore, you’re not Jewish”. Is it going to be a thing that I will have to defend myself constantly? Will I have to always wonder if I meet a group of Jewish people that I will be slapped with the “you’re not really Jewish”?

And converts, how do/did you deal with this issue?