Unimodality of traces of circular-fence matrices

Prove that for every sequence of positive integers a_1, ..., a_{2m}, the trace of the q-deformed matrix M_q(a_1, ..., a_{2m}) is unimodal, except when the sequence is (1,k,1,k) or (k,1,k,1) for some k; establish unimodality in these exceptional cases when m=2.

Background

For a sequence of positive integers, the paper associates a q-deformed matrix M_q(a_1, ..., a_{2m}) and identifies its trace with the rank polynomial of a circular fence poset. The cited conjecture concerns the coefficient sequence of this trace polynomial and asserts unimodality apart from two explicitly described alternating patterns.

This conjecture is introduced as a conjecture of Kantarcı Oğuz and is subsequently used to motivate a related conjecture for normalized Jones polynomials of rational links and, equivalently, for the polynomials I_a(q) arising from left q-deformed rational numbers.