Unimodality of traces of q-deformed modular-group matrices

Prove that for every sequence of positive integers a_1,\ldots,a_{2m}, the trace of the q-deformed modular-group matrix M_q(a_1,\ldots,a_{2m}) is unimodal, except when (a_1,a_2,\ldots,a_{2m})=(1,k,1,k) or (k,1,k,1) for some positive integer k, with the exceptional cases also satisfying unimodality when m=2.

Background

For a sequence of positive integers a_1,\ldots,a_{2m}, the paper considers the matrix M_q(a_1,\ldots,a_{2m}) constructed from the q-deformed modular-group generators. The trace of this matrix is identified with the rank polynomial of a circular fence poset, connecting the algebraic question to the combinatorics of poset rank polynomials.

The conjecture is attributed to Kantarcı Oğuz and concerns the coefficient sequence of the trace polynomial. It asserts unimodality in general, allowing only two specified alternating exceptional families; the paper notes that the cases with m=2 are unimodal even for those families.