  Recently, Nunge studied Eulerian polynomials on segmented permutations,
namely \emph{generalized Eulerian polynomials}, and further asked whether their
coefficients form unimodal sequences. In this paper, we prove the stability of
the generalized Eulerian polynomials and hence confirm Nunge's conjecture. Our
proof is based on Br\"and\'en's stable multivariate Eulerian polynomials. By
acting on Br\"and\'en's polynomials with a stability-preserving linear
operator, we get a multivariate refinement of the generalized Eulerian
polynomials. To prove Nunge's conjecture, we also develop a general approach to
obtain generalized Sturm sequences from bivariate stable polynomials.
