Kantarcı Oğuz's unimodality conjecture for traces of q-deformed matrices

For a sequence of positive integers $a_1,\ldots,a_{2m}$, let $M(a_1,\ldots,a_{2m})$ be the corresponding matrix and call a polynomial $f(q)=\sum_{i=0}^n a_iq^i$ unimodal if $a_0\le\cdots\le a_j\ge\cdots\ge a_n$ for some $j$ with $0\le j\le n$. Kantarcı Oğuz's conjecture. For any sequence $a_1,\ldots,a_{2m}>0$, the trace $\operatorname{tr}(M(a_1,\ldots,a_{2m}))$ is unimodal except when $(a_1,a_2,\ldots,a_{2m})=(1,k,1,k)$ or $(k,1,k,1)$ for some $k$; the case $m=2$ is unimodal even in these exceptional cases. This conjecture concerns the coefficient shape of traces related to circular fence posets and would imply corresponding unimodality statements for normalized Jones polynomials of rational links. Its resolution is not given in the supplied text.