A potential analogue of Schinzel's hypothesis for polynomials with   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 of the finite field Fq 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. 
Andreas O. Bender, Olivier Wittenberg