Uniformization over finite fields

I just asked this question on MathOverflow.  It's basically idle curiosity, but I've now been idly curious about this for several years, so I figured I might as well ask it publicly.  Please let me know if you have any thoughts!

In short, the question is:  Do every two smooth projective curves over \(\overline{\mathbb{F}_q}\) of genus at least \(2\) have a finite etale cover in common?

See the question for more details and remarks.