Pointless, non-singular , absolutely irreducible plane curves over finite fields

We think the following is true:

For all sufficiently large primes $$p$$ and all natural $$g \ge 1$$, there exists plane curve $$f(x,y)=0$$ over $$\mathbb{F}_p$$ which is non-singular, absolutely irreducible, of genus $$g$$ and it doesn’t have any rational points.

Is it true?

Is it known?