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?