Unimodality of the generalized k-regular partition polynomials

Prove that, for every integer k 2 and every choice of non-negative integers n_k,n_{k-1},,n_1, the polynomial b(n_k,,n_1) constructed in Theorem 3 is unimodal.

Background

Theorem 3 constructs polynomials b(n_k,n_{k-1},,n_1) arising as coefficients in a generating function for k-regular partitions. The paper notes that empirical evidence indicates these polynomials are always unimodal, extending the observed unimodality of the small cases displayed earlier and differing from known unimodality results for coefficients of Bessel polynomials. The authors explicitly leave this assertion as a conjecture for all k 2 and all non-negative parameter values.