Trace counterexample classification conjecture

Let $c_1,c_2,\ldots,c_k$ be integers with $c_i\geq 2$, and let $M_q(c_1,c_2,\ldots,c_k)$ be the associated matrix. Two sequences are considered equivalent when one is obtained from the other by a cyclic shift. Trace counterexample classification conjecture. Up to cyclic shifts of $\llbracket c_1,c_2,\ldots,c_k\rrbracket$, the examples listed in the paper's counterexample corollary are the only counterexamples to unimodality for $\mathrm{tr}(M_q(c_1,c_2,\ldots,c_k))$. The claim seeks a complete classification of failures of unimodality under the condition $c_i\geq2$, building on the explicit counterexamples displayed immediately beforehand.