Maybe that's the wrong question. That question assumes that there's a zero-sum tension between these two ideas: the more "yourself" you are, then the less "relevant" you are. According to this idea, you have to attempt to cash in authenticity to gain acceptance.
But really, the you that comes through, the you that desperately wants to be liked, that's the real you. You can wear a mask to hide your identity, but you will still be known as an actor or a bandit. Who else wears masks?
What if there was a different way to be relevant? Hack your way into the jungle of your own story. Search for a clearing large enough to accommodate the people with whom you're longing to connect. Where is there space for them in your story? What experiences do you share? Fears? Dreams? Struggles? Questions? Hopes? Joys?
Throw down a blanket and send out the invitations. Each of us have clearings in the jumbled jungle of our life-story. If you can invite people into those clear spaces, you can be relevant without being fake.
I don't know of any other way.