Leclere–Morier-Genoud trace unimodality conjecture

Let $c_1,c_2,\ldots,c_k$ be integers, and let $M_q(c_1,c_2,\ldots,c_k)$ be the associated matrix in $\mathrm{PSL}_q(2,\mathbb{Z})$. Leclere–Morier-Genoud trace unimodality conjecture. For any sequence $c_1,c_2,\ldots,c_k$ of integers, the trace $\mathrm{tr}(M_q(c_1,c_2,\ldots,c_k))$ is a polynomial in $\mathbb{Z}[q,q^{-1}]$ with unimodal coefficients. The paper states this conjecture as an analogue of the unimodality question for circular rank polynomials, but later gives evidence that the cited conjecture does not hold.