A potential analogue of Schinzel's hypothesis for polynomialswith coefficients in Fq[t]

The Schinzel hypothesis essentially claims that finitely many irreducible polynomials in one variable over ℤ simultaneously assume infinitely many prime values unless there is an obvious reason why this is impossible. We prove that under a restriction on the characteristic and a smoothness assumption, finitely many irreducible polynomials in one variable over the ring Fq[t] assume simultaneous prime values after a sufficiently large extension of the field of constants.